If inverse of matrix $\begin{bmatrix} 7 & -3 & -3 \\ -1 & 1 & 0 \\ -1 & 0 & 1 \end{bmatrix}$ is the matrix $\begin{bmatrix} 1 & 3 & 3 \\ 1 & \lambda & 3 \\ 1 & 3 & 4 \end{bmatrix}$, then value of $\lambda$ is:
(A) -4
(B) 1
(C) 3
(D) 4
CBSE CLASS 12 • MATHEMATICS • CHAPTER 03
Matrices
352 distinct questions across 495 board paper set appearances (2022–2026).
Find the matrix $A^2$, where $A = [a_{ij}]$ is a $2 \times 2$ matrix whose elements are given by $a_{ij} = \text{maximum} (i, j) - \text{minimum} (i, j) :$ (A) $\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$ (B) $\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$ (C) $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{b...
If $A$ is a square matrix of order 3 such that the value of $|\text{adj } A| = 8$, then the value of $|A^T|$ is:
(A) $\sqrt{2}$
(B) $-\sqrt{2}$
(C) 8
(D) $2\sqrt{2}$
Which of the following properties is/are true for two matrices of suitable orders ?
(i) $(A + B)' = A' + B'$
(ii) $(A - B)' = B' - A'$
(iii) $(AB)' = A'B'$
(iv) $(kAB)' = kB'A'$ ($k$ is a scalar)
(A) (i) only
(B) (i), (ii) and (iii)
(C) (i) and (ii)
(D) (i) and (iv)
If $\Delta_1 = \begin{vmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{vmatrix}$ and $\Delta_2 = \begin{vmatrix} 0 & 2 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 6 \end{vmatrix}$, then
(A) $\Delta_1 = 2\Delta_2$
(B) $\Delta_2 = -2\Delta_1$
(C) $\Delta_1 = \Delta_2$
(D) $\Delta_2 = -\Delta_1$
If $A = \begin{bmatrix} 1 & a & b \\ -1 & 2 & c \\ 0 & 5 & 3 \end{bmatrix}$ is a symmetric matrix, then the value of $3a + b + c$ is
(A) $2$
(B) $6$
(C) $4$
(D) $0$
For a square matrix $A$, $(3A)^{-1} =$
(A) $3A^{-1}$
(B) $9A^{-1}$
(C) $\frac{1}{3}A^{-1}$
(D) $\frac{1}{9}A^{-1}$
If $\begin{vmatrix} -1 & -2 & 5 \\ -2 & a & -1 \\ 0 & 4 & 2a \end{vmatrix} = -86$, then the sum of all possible values of $a$ is
(A) $4$
(B) $5$
(C) $-4$
(D) $9$
Assertion (A) : If A and B are two square matrices such that AB and BA are defined, then it is not necessary that AB = BA.
Reason (R) : Product of two diagonal matrices of same order is commutative.
Let $A = \begin{bmatrix} 0 & -3 & 4 \\ 1 & 0 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} -3 & 0 & 1 \\ 2 & 4 & 0 \end{bmatrix}$. If $A + B + C = O$, then matrix $C$ is :
(A) $\begin{bmatrix} -3 & -3 & 5 \\ 3 & 4 & 2 \end{bmatrix}$
(B) $\begin{bmatrix} 3 & 3 & 5 \\ -3 & -4 & -2 \end{bmatrix}$
(C) $\b...
If $A = \begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{bmatrix}$ satisfies $A' + A = I$, then the value of $\theta$ is :
(A) $\frac{\pi}{6}$
(B) $\frac{\pi}{3}$
(C) $\frac{2\pi}{3}$
(D) $\frac{5\pi}{6}$
If $\begin{vmatrix} 3x & x+2 \\ 3(x-1) & x-2 \end{vmatrix} = \begin{vmatrix} 1 & 5 \\ 3 & 3 \end{vmatrix}$, then the value of $x$ is :
(A) $2$
(B) $\frac{1}{3}$
(C) $5$
(D) $3$
If $A$ and $B$ are square matrices of order $3$ such that $|A| = 4$ and $|B| = 7$, then $|2AB|$ is :
(A) $112$
(B) $56$
(C) $28$
(D) $224$
If $A = \begin{bmatrix} 1 & -2 & 3 \\ 3 & 2 & -1 \end{bmatrix}$, $B = \begin{bmatrix} 2 & 3 \\ -1 & 2 \\ 4 & -5 \end{bmatrix}$ and $C = [C_{ij}] = AB$, then $C_{12}$ is :
(A) $22$
(B) $-16$
(C) $4$
(D) $0$
(a) Express the matrix $A = \begin{bmatrix} 2 & 4 \\ 5 & 6 \end{bmatrix}$ as the sum of a symmetric matrix and a skew symmetric matrix.
OR
(b) Two lines are represented by equations $5x - 7y - 2 = 0$ and $7x - 5y - 3 = 0$ in the $xy$-plane. Using matrix method, find their point of intersection.
If $A = \begin{bmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$, then $A^{-1}$ is
(A) $\begin{bmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{bmatrix}$
(B) $\begin{bmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{bmatrix}$
(C) $\begin{bmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & ...
If A is a $3 \times 3$ invertible matrix, show that for any scalar $k \neq 0$, $(kA)^{-1} = \frac{1}{k} A^{-1}$. Hence calculate $(3A)^{-1}$, where $A = \begin{bmatrix} 2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2 \end{bmatrix}$.
If $A = \begin{bmatrix} 5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5 \end{bmatrix}$, then $A^3$ is :
(A) $3 \begin{bmatrix} 5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5 \end{bmatrix}$
(B) $\begin{bmatrix} 125 & 0 & 0 \\ 0 & 125 & 0 \\ 0 & 0 & 125 \end{bmatrix}$
(C) $\begin{bmatrix} 15 & 0 & 0 \\ 0 & 15 & 0 \\ 0 & 0 &...
If $\begin{vmatrix} 2x & 5 \\ 12 & x \end{vmatrix} = \begin{vmatrix} 6 & -5 \\ 4 & 3 \end{vmatrix}$, then the value of x is :
(A) 3
(B) 7
(C) $\pm 7$
(D) $\pm 3$
Let both AB' and B'A be defined for matrices A and B. If order of A is $n \times m$, then the order of B is :
(A) $n \times n$
(B) $n \times m$
(C) $m \times m$
(D) $m \times n$
Sum of two skew-symmetric matrices of same order is always a/an :
(A) skew-symmetric matrix
(B) symmetric matrix
(C) null matrix
(D) identity matrix
If the sum of all the elements of a $3 \times 3$ scalar matrix is $9$, then the product of all its elements is :
(A) $0$
(B) $9$
(C) $27$
(D) $729$
If $\begin{vmatrix} -a & b & c \\ a & -b & c \\ a & b & -c \end{vmatrix} = kabc$, then the value of $k$ is :
(A) $0$
(B) $1$
(C) $2$
(D) $4$
Assertion (A) : For any symmetric matrix A, $B'AB$ is a skew-symmetric matrix. Reason (R) : A square matrix P is skew-symmetric if $P' = -P$.
If A = [a_{ij}] is an identity matrix, then which of the following is true ?
(A) a_{ij} = \begin{cases} 0, & \text{if } i = j \\ 1, & \text{if } i \neq j \end{cases}
(B) a_{ij} = 1, \forall i, j
(C) a_{ij} = 0, \forall i, j
(D) a_{ij} = \begin{cases} 0, & \text{if } i \neq j \\ 1, & \text{if } i = j...
Let $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ be a square matrix such that $\text{adj } A = A$. Then, $(a + b + c + d)$ is equal to :
(A) $2a$
(B) $2b$
(C) $2c$
(D) $0$
If A = $\begin{bmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \end{bmatrix}$, then $A^{-1}$ is :
(A) $\begin{bmatrix} \frac{1}{2} & 0 & 0 \\ 0 & \frac{1}{3} & 0 \\ 0 & 0 & \frac{1}{5} \end{bmatrix}$
(B) $30 \begin{bmatrix} \frac{1}{2} & 0 & 0 \\ 0 & \frac{1}{3} & 0 \\ 0 & 0 & \frac{1}{5} \end{bmatrix}...
Assertion (A) : Every scalar matrix is a diagonal matrix. Reason (R) : In a diagonal matrix, all the diagonal elements are 0.
If $\begin{bmatrix} a & c & 0 \\ b & d & 0 \\ 0 & 0 & 5 \end{bmatrix}$ is a scalar matrix, then the value of $a + 2b + 3c + 4d$ is :
(A) $0$
(B) $5$
(C) $10$
(D) $25$
Given that $A^{-1} = \frac{1}{7} \begin{bmatrix} 2 & 1 \\ -3 & 2 \end{bmatrix}$, matrix A is : (A) $7 \begin{bmatrix} 2 & -1 \\ 3 & 2 \end{bmatrix}$ (B) $\begin{bmatrix} 2 & -1 \\ 3 & 2 \end{bmatrix}$ (C) $\frac{1}{7} \begin{bmatrix} 2 & -1 \\ 3 & 2 \end{bmatrix}$ (D) $\frac{1}{49} \begin{bmatrix} 2...
(b) Find the product of the matrices $\begin{bmatrix} 1 & 2 & -3 \\ 2 & 3 & 2 \\ 3 & -3 & -4 \end{bmatrix} \begin{bmatrix} -6 & 17 & 13 \\ 14 & 5 & -8 \\ -15 & 9 & -1 \end{bmatrix}$ and hence solve the system of linear equations :
$x + 2y - 3z = -4$
$2x + 3y + 2z = 2$
$3x - 3y - 4z = 11$
(a) If $A = \begin{bmatrix} 1 & 2 & -3 \\ 2 & 0 & -3 \\ 1 & 2 & 0 \end{bmatrix}$, then find $A^{-1}$ and hence solve the following system of equations : $x + 2y - 3z = 1$, $2x - 3z = 2$, $x + 2y = 3$
An architect is developing a plot of land for a commercial complex. When asked about the dimensions of the plot, he said that if the length is decreased by $25\text{ m}$ and the breadth is increased by $25\text{ m}$, then its area increases by $625\text{ m}^2$. If the length is decreased by $20\text...
If for a square matrix A, $A^2 - 3A + I = O$ and $A^{-1} = xA + yI$, then the value of $x + y$ is : (a) $-2$ (b) $2$ (c) $3$ (d) $-3$
If $\begin{bmatrix} 2 & 0 \\ 5 & 4 \end{bmatrix} = P + Q$, where P is a symmetric and Q is a skew symmetric matrix, then Q is equal to
(A) $\begin{bmatrix} 2 & 5/2 \\ 5/2 & 4 \end{bmatrix}$
(B) $\begin{bmatrix} 0 & -5/2 \\ 5/2 & 0 \end{bmatrix}$
(C) $\begin{bmatrix} 0 & 5/2 \\ -5/2 & 0 \end{bmatrix}...
If $\begin{bmatrix} 1 & 2 & 1 \\ 2 & 3 & 1 \\ 3 & a & 1 \end{bmatrix}$ is non-singular matrix and $a \in A$, then the set A is
(A) $\mathbb{R}$
(B) $\{0\}$
(C) $\{4\}$
(D) $\mathbb{R} - \{4\}$
If $|A| = |kA|$, where A is a square matrix of order 2, then sum of all possible values of k is
(A) 1
(B) $-1$
(C) 2
(D) 0
If a matrix $A = \begin{bmatrix} 1 & 2 & 3 \end{bmatrix}$, then the matrix $AA'$ (where $A'$ is the transpose of $A$) is :
(a) $14$
(b) $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix}$
(c) $\begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 1 \\ 3 & 1 & 2 \end{bmatrix}$
(d) $[14]$
If matrix $A = \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix}$ and $A^2 = kA$, then the value of $k$ is :
(a) $1$
(b) $-2$
(c) $2$
(d) $-1$
It is given that $X \begin{bmatrix} 3 & 2 \\ 1 & -1 \end{bmatrix} = \begin{bmatrix} 4 & 1 \\ 2 & 3 \end{bmatrix}$. Then matrix $X$ is :
(a) $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$
(b) $\begin{bmatrix} 0 & -1 \\ 1 & 1 \end{bmatrix}$
(c) $\begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}$
(d) $\beg...
If $A$ and $B$ are square matrices of same order, then which of the following statements is/are always true ?
(i) $(A + B) (A - B) = A^2 - B^2$
(ii) $AB = BA$
(iii) $(A + B)^2 = A^2 + AB + BA + B^2$
(iv) $AB = 0 \Rightarrow A = 0$ or $B = 0$
(A) Only (i) and (iii)
(B) Only (ii) and (iii)
(C) Only (i...
(a) If $\text{P} = \begin{bmatrix} 1 & -1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2 \end{bmatrix}$ and $\text{Q} = \begin{bmatrix} 2 & 2 & -4 \\ -4 & 2 & -4 \\ 2 & -1 & 5 \end{bmatrix}$, find $(\text{QP})$ and hence solve the following system of equations using matrices :\n$x - y = 3$, $2x + 3y + 4z = 17$, $y +...
If A and B are skew-symmetric matrices of same order, then $AB' + BA'$ is a/an :
(A) symmetric matrix
(B) skew-symmetric matrix
(C) null matrix
(D) identity matrix
A total of $60$ vouchers were given on the day. The number of ₹ $1,000$ vouchers added to $3$ times the number of ₹ $500$ vouchers, gives $100$. Express the given information as a system of linear equations in two variables. Hence, find the number of vouchers of each type by matrix method.
OR
(b) ...
Write the equations representing the various dimensions and express them as the matrix equation $AX = B$.
If $A$ is a square matrix, then which of the following is \textit{not} true ?
(A) $A + A'$ is a symmetric matrix
(B) $AA'$ and $A'A$ are symmetric matrices
(C) $A + A'$ is a skew symmetric matrix
(D) $A - A'$ is a skew symmetric matrix
The value of the determinant $\begin{vmatrix} \cos 75^{\circ} & \sin 75^{\circ} \\ \sin 15^{\circ} & \cos 15^{\circ} \end{vmatrix}$ is :
(A) $1$
(B) zero
(C) $\frac{1}{2}$
(D) $\frac{\sqrt{3}}{2}$
If a matrix has $36$ elements, the number of possible orders it can have, is :
(A) $13$
(B) $3$
(C) $5$
(D) $9$
Assertion (A) : For matrix $A = \begin{bmatrix} 1 & \cos \theta & 1 \\ -\cos \theta & 1 & \cos \theta \\ -1 & -\cos \theta & 1 \end{bmatrix}$, where $\theta \in [0, 2\pi]$, $|A| \in [2, 4]$.
Reason (R) : $\cos \theta \in [-1, 1], \forall \theta \in [0, 2\pi]$.
If $A$ and $B$ are invertible matrices, then which of the following is not correct ?
(A) $(A + B)^{-1} = B^{-1} + A^{-1}$
(B) $(AB)^{-1} = B^{-1}A^{-1}$
(C) $\text{adj } (A) = | A | A^{-1}$
(D) $| A |^{-1} = | A^{-1} |$
A school wants to allocate students into three clubs : Sports, Music and Drama, under following conditions :
• The number of students in Sports club should be equal to the sum of the number of students in Music and Drama club.
• The number of students in Music club should be 20 more than half the nu...
Which of the following can be both a symmetric and skew-symmetric matrix ?
(A) Unit Matrix
(B) Diagonal Matrix
(C) Null Matrix
(D) Row Matrix
Four friends Abhay, Bina, Chhaya and Devesh were asked to simplify $4 \text{AB} + 3(\text{AB} + \text{BA}) - 4 \text{BA}$, where $A$ and $B$ are both matrices of order $2 \times 2$. It is known that $A \neq B \neq I$ and $A^{-1} \neq B$.
Their answers are given as :
Abhay : $6 \text{AB}$
Bina : $7 \...
If $A$ and $B$ are square matrices of order $m$ such that $A^2 - B^2 = (A - B) (A + B)$, then which of the following is always correct ?
(A) $A = B$
(B) $AB = BA$
(C) $A = 0$ or $B = 0$
(D) $A = I$ or $B = I$
(a) Given $A = \begin{bmatrix} -4 & 4 & 4 \\ -7 & 1 & 3 \\ 5 & -3 & -1 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & -1 & 1 \\ 1 & -2 & -2 \\ 2 & 1 & 3 \end{bmatrix}$, find $AB$. Hence, solve the system of linear equations:
$x - y + z = 4$
$x - 2y - 2z = 9$
$2x + y + 3z = 1$
OR
(b) If $A = \begin{bmat...
If $A$ and $B$ are square matrices of same order such that $AB = A$ and $BA = B$, then $A^2 + B^2$ is equal to :
(A) $A + B$
(B) $BA$
(C) $2 (A + B)$
(D) $2BA$
The matrix $\begin{bmatrix} 0 & 1 & -2 \\ -1 & 0 & -7 \\ 2 & 7 & 0 \end{bmatrix}$ is a : (A) diagonal matrix (B) symmetric matrix (C) skew symmetric matrix (D) scalar matrix
If A = $[a_{ij}]$ is a $3 \times 3$ diagonal matrix such that $a_{11} = 1$, $a_{22} = 5$ and $a_{33} = -2$, then $|A|$ is :
(A) 0
(B) -10
(C) 10
(D) 1
Three students, Neha, Rani and Sam go to a market to purchase stationery items. Neha buys 4 pens, 3 notepads and 2 erasers and pays ₹ 60. Rani buys 2 pens, 4 notepads and 6 erasers for ₹ 90. Sam pays ₹ 70 for 6 pens, 2 notepads and 3 erasers. Based upon the above information, answer the following qu...
(a) Let $2x + 5y - 1 = 0$ and $3x + 2y - 7 = 0$ represent the equations of two lines on which the ants are moving on the ground. Using matrix method, find a point common to the paths of the ants.
OR
(b) A shopkeeper sells $50$ Chemistry, $60$ Physics and $35$ Maths books on day I and sells $40$ Ch...
If A and B are two square matrices each of order 3 with $|A| = 3$ and $|B| = 5$, then $|2AB|$ is :
(A) 30
(B) 120
(C) 15
(D) 225
The matrix A = $\begin{bmatrix} \sqrt{3} & 0 & 0 \\ 0 & \sqrt{2} & 0 \\ 0 & 0 & \sqrt{5} \end{bmatrix}$ is a/an :
(A) scalar matrix
(B) identity matrix
(C) null matrix
(D) symmetric matrix
What is the total number of possible matrices of order $3 \times 3$ with each entry as $\sqrt{2}$ or $\sqrt{3}$ ?
(A) 9
(B) 512
(C) 615
(D) 64
If $F(x) = \begin{bmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{bmatrix}$ and $[F(x)]^2 = F(kx)$, then the value of k is : (A) 1 (B) 2 (C) 0 (D) -2
(a) Solve the following system of equations, using matrices : $\frac{2}{x} + \frac{3}{y} + \frac{10}{z} = 4, \frac{4}{x} - \frac{6}{y} + \frac{5}{z} = 1, \frac{6}{x} + \frac{9}{y} - \frac{20}{z} = 2$ where $x, y, z \neq 0$ OR (b) If $A = \begin{bmatrix} 1 & \cot x \\ -\cot x & 1 \end{bmatrix}$, show...
If A and B are two skew symmetric matrices, then (AB + BA) is :
(A) a skew symmetric matrix
(B) a symmetric matrix
(C) a null matrix
(D) an identity matrix
If $\begin{vmatrix} 1 & 3 & 1 \\ k & 0 & 1 \\ 0 & 0 & 1 \end{vmatrix} = \pm 6$, then the value of k is :
(A) 2
(B) -2
(C) $\pm 2$
(D) $\mp 2$
A scholarship is a sum of money provided to a student to help him or her pay for education. Some students are granted scholarships based on their academic achievements, while others are rewarded based on their financial needs. Every year a school offers scholarships to girl children and meritorious ...
If $A = \begin{bmatrix} 2 & 1 \\ -4 & -2 \end{bmatrix}$, then the value of $I - A + A^2 - A^3 + \dots$ is :
(A) $\begin{bmatrix} -1 & -1 \\ 4 & 3 \end{bmatrix}$
(B) $\begin{bmatrix} 3 & 1 \\ -4 & -1 \end{bmatrix}$
(C) $\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$
(D) $\begin{bmatrix} 1 & 0 \\ 0 & 1...
If $A = \begin{bmatrix} a & c & -1 \\ b & 0 & 5 \\ 1 & -5 & 0 \end{bmatrix}$ is a skew-symmetric matrix, then the value of $2a - (b + c)$ is:
(A) 0
(B) 1
(C) -10
(D) 10
If $\begin{bmatrix} x & 2 & 0 \end{bmatrix} \begin{bmatrix} 5 \\ -1 \\ x \end{bmatrix} = \begin{bmatrix} 3 & 1 \end{bmatrix} \begin{bmatrix} -2 \\ x \end{bmatrix}$, then value of $x$ is:
(A) -1
(B) 0
(C) 1
(D) 2
(a) If A = $\begin{bmatrix} 1 & 2 & -3 \\ 2 & 0 & -3 \\ 1 & 2 & 0 \end{bmatrix}$, then find $A^{-1}$ and hence solve the following system of equations :
$x + 2y - 3z = 1$
$2x - 3z = 2$
$x + 2y = 3$
OR
If $A$ is a square matrix of order 2 and $|A| = -2$, then value of $|5A'|$ is :
(A) $-50$
(B) $-10$
(C) $10$
(D) $50$
The product of matrix $P$ and $Q$ is equal to a diagonal matrix. If the order of matrix $Q$ is $3 \times 2$, then order of matrix $P$ is :
(A) $2 \times 2$
(B) $3 \times 3$
(C) $2 \times 3$
(D) $3 \times 2$
For the matrix A = $\begin{bmatrix} 2 & -1 & 1 \\ \lambda & 2 & 0 \\ 1 & -2 & 3 \end{bmatrix}$ to be invertible, the value of $\lambda$ is :
(A) $0$
(B) $10$
(C) $\mathbb{R} - \{10\}$
(D) $\mathbb{R} - \{-10\}$
If $A = \begin{bmatrix} x & 0 \\ 1 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 4 & 0 \\ -1 & 1 \end{bmatrix}$, then value of $x$ for which $A^2 = B$ is :
(A) $-2$
(B) $2$
(C) $2$ or $-2$
(D) $4$
Let $f(x) = \begin{vmatrix} x^2 & \sin x \\ p & -1 \end{vmatrix}$, where $p$ is a constant. The value of $p$ for which $f'(0) = 1$ is :
(A) $\mathbb{R}$
(B) $1$
(C) $0$
(D) $-1$
For two matrices $A$ and $B$, given that $A^{-1} = \frac{1}{4}B$, then inverse of $(4A)$ is : (A) $4B$ (B) $B$ (C) $\frac{1}{4}B$ (D) $\frac{1}{16}B$
If X, Y and XY are matrices of order $2 \times 3$, $m \times n$ and $2 \times 5$ respectively, then number of elements in matrix Y is : (A) 6 (B) 10 (C) 15 (D) 35
Let A be a $3 \times 3$ matrix such that $|\text{adj } A| = 64$. Then $|A|$ is equal to : (a) $8$ only (b) $-8$ only (c) $64$ (d) $8$ or $-8$
If A = $\begin{bmatrix} 3 & 4 \\ 5 & 2 \end{bmatrix}$ and 2A + B is a null matrix, then B is equal to : (a) $\begin{bmatrix} 6 & 8 \\ 10 & 4 \end{bmatrix}$ (b) $\begin{bmatrix} -6 & -8 \\ -10 & -4 \end{bmatrix}$ (c) $\begin{bmatrix} 5 & 8 \\ 10 & 3 \end{bmatrix}$ (d) $\begin{bmatrix} -5 & -8 \\ -10 ...
If $A = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$ and $(3I + 4A)(3I - 4A) = x^2I$, then the value(s) x is/are :
(a) $\pm \sqrt{7}$
(b) 0
(c) $\pm 5$
(d) 25
(a) If $A = \begin{bmatrix} 1 & 2 & -2 \\ -1 & 3 & 0 \\ 0 & -2 & 1 \end{bmatrix}$ and $B^{-1} = \begin{bmatrix} 3 & -1 & 1 \\ -15 & 6 & -5 \\ 5 & -2 & 2 \end{bmatrix}$, find $(AB)^{-1}$.
OR (b) Solve the following system of equations by matrix method : $x + 2y + 3z = 6$ $2x - y + z = 2$ $3x + 2y - 2z = 3$
(a) If A = $\begin{bmatrix} -3 & -2 & -4 \\ 2 & 1 & 2 \\ 2 & 1 & 3 \end{bmatrix}$, B = $\begin{bmatrix} 1 & 2 & 0 \\ -2 & -1 & -2 \\ 0 & -1 & 1 \end{bmatrix}$, then find AB and use it to solve the following system of equations: x - 2y = 3, 2x - y - z = 2, -2y + z = 3
If $(a, b), (c, d)$ and $(e, f)$ are the vertices of $\Delta ABC$ and $\Delta$ denotes the area of $\Delta ABC$, then $\begin{vmatrix} a & c & e \\ b & d & f \\ 1 & 1 & 1 \end{vmatrix}^2$ is equal to
(A) $2\Delta^2$
(B) $4\Delta^2$
(C) $2\Delta$
(D) $4\Delta$
For what value of $x \in \left[ 0, \frac{\pi}{2} \right]$, is $A + A' = \sqrt{3} I$, where
$A = \begin{bmatrix} \cos x & \sin x \\ -\sin x & \cos x \end{bmatrix}$?
(a) $\frac{\pi}{3}$
(b) $\frac{\pi}{6}$
(c) 0
(d) $\frac{\pi}{2}$
Let A be the area of a triangle having vertices $(x_1, y_1)$, $(x_2, y_2)$ and $(x_3, y_3)$. Which of the following is correct ?
(a) $\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} = \pm A$
(b) $\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatr...
The product $\begin{bmatrix} a & b \\ -b & a \end{bmatrix} \begin{bmatrix} a & -b \\ b & a \end{bmatrix}$ is equal to :
(a) $\begin{bmatrix} a^2+b^2 & 0 \\ 0 & a^2+b^2 \end{bmatrix}$
(b) $\begin{bmatrix} (a+b)^2 & 0 \\ (a+b)^2 & 0 \end{bmatrix}$
(c) $\begin{bmatrix} a^2+b^2 & 0 \\ a^2+b^2 & 0 \end{b...
If $A$ is a square matrix and $A^2 = A$, then $(I + A)^2 - 3A$ is equal to :
(a) $I$
(b) $A$
(c) $2A$
(d) $3I$
Find the inverse of the matrix $A = \begin{bmatrix} 1 & -1 & 2 \\ 0 & 2 & -3 \\ 3 & -2 & 4 \end{bmatrix}$. Using the inverse, $A^{-1}$, solve the system of linear equations
$x - y + 2z = 1$; $2y - 3z = 1$; $3x - 2y + 4z = 3$.
If $x \begin{bmatrix} 1 \\ 2 \end{bmatrix} + y \begin{bmatrix} 2 \\ 5 \end{bmatrix} = \begin{bmatrix} 4 \\ 9 \end{bmatrix}$, then :
(a) $x = 1, y = 2$
(b) $x = 2, y = 1$
(c) $x = 1, y = -1$
(d) $x = 3, y = 2$
If $A = \begin{bmatrix} 1 & 0 \\ 2 & 1 \end{bmatrix}$, $B = \begin{bmatrix} x & 0 \\ 1 & 1 \end{bmatrix}$ and $A = B^2$, then $x$ equals
(a) $\pm 1$
(b) $-1$
(c) 1
(d) 2
For which value of $x$, are the determinants $\begin{vmatrix} 2x & -3 \\ 5 & x \end{vmatrix}$ and $\begin{vmatrix} 10 & 1 \\ -3 & 2 \end{vmatrix}$ equal?
(a) $\pm 3$
(b) $-3$
(c) $\pm 2$
(d) $2$
The value of the cofactor of the element of second row and third column in the matrix $\begin{bmatrix} 4 & 3 & 2 \\ 2 & -1 & 0 \\ 1 & 2 & 3 \end{bmatrix}$ is :
(a) $5$
(b) $-5$
(c) $-11$
(d) $11$
Which of the following cannot be the order of a row-matrix ?
(A) $2 \ imes 1$
(B) $1 \ imes 2$
(C) $1 \ imes 1$
(D) $1 \ imes n$
One of the values of $x$ for which $\begin{vmatrix} \cos x & \sin x \\ -\cos x & \sin x \end{vmatrix} = 1$ is
(A) $0$
(B) $\\frac{\pi}{4}$
(C) $\\frac{\pi}{3}$
(D) $\\frac{\pi}{2}$
If $A = \begin{bmatrix} 0 & 2 & 1 \\ -2 & -1 & -2 \\ 1 & -1 & 0 \end{bmatrix}$, find $A^{-1}$ and use it to solve the following system of equations : $-2y + z = 7$, $2x - y - z = 8$, $x - 2y = 10$
If $\begin{bmatrix} 3 & -1 & \sin 3x \\ -7 & 4 & \cos 2x \\ -11 & 7 & 2 \end{bmatrix}$ is a singular matrix, then find all values of $x$ where $x \in \left[0, \\frac{\pi}{2}\right]$.
Which of the following cannot be an order of a column-matrix ?
(A) $1 \times 2$
(B) $2 \times 1$
(C) $1 \times 1$
(D) $m \times 1$, where $m \in \mathbb{N}$
One of the values of $x$ for which $\begin{vmatrix} \cos x & \sin x \\ -\cos x & \sin x \end{vmatrix} = 1$ is
(A) 0
(B) $\frac{\pi}{4}$
(C) $\frac{\pi}{3}$
(D) $\frac{\pi}{2}$
If $A$ and $B$ are symmetric matrices of same order, then which of the following matrices is a skew-symmetric matrix ?
(A) $A^2 + B^2$
(B) $A^2 - B^2$
(C) $AB + BA$
(D) $AB - BA$
(a) If $A = \begin{bmatrix} 0 & 2 & 1 \\ -2 & -1 & -2 \\ 1 & -1 & 0 \end{bmatrix}$, find $A^{-1}$ and use it to solve the following system of equations : $-2y + z = 7$, $2x - y - z = 8$, $x - 2y = 10$
OR
(b) If $\begin{bmatrix} 3 & -1 & \sin 3x \\ -7 & 4 & \cos 2x \\ -11 & 7 & 2 \end{bmatrix}$ is ...
If $A = [a_{ij}]_{3 \times 3}$ is a scalar matrix then which of the following must be true ?
(A) A must be a symmetric matrix.
(B) A must be a skew-symmetric matrix.
(C) A must be an identity matrix.
(D) A must be a null matrix.
One of the values of $x$ for which $\begin{vmatrix} \cos x & \\ -\cos x & \sin x \end{vmatrix} = 1$ is
(A) $0$
(B) $\frac{\pi}{4}$
(C) $\frac{\pi}{3}$
(D) $\frac{\pi}{2}$
If A and B are symmetric matrices of same order, then $(AB - BA)$ is a
(A) Zero matrix
(B) Identity matrix
(C) Symmetric matrix
(D) Skew symmetric matrix
(a) If $\text{A} = \begin{bmatrix} 0 & 2 & 1 \\ -2 & -1 & -2 \\ 1 & -1 & 0 \end{bmatrix}$, find $\text{A}^{-1}$ and use it to solve the following system of equations :<br>$-2y + z = 7$, $2x - y - z = 8$, $x - 2y = 10$<br>OR
If $\begin{vmatrix} 3 & -1 & \sin 3x \\ -7 & 4 & \cos 2x \\ -11 & 7 & 2 \end{vmatrix}$ is a singular matrix, then find all values of $x$ where $x \in \left[0, \frac{\pi}{2}\right]$.
If $A = \begin{bmatrix} \cos x & -\sin x \\ \sin x & \cos x \end{bmatrix}$ and $A + A' = I$, then the value of $x \in \left[0, \frac{\pi}{2}\right]$ is
(A) $0$
(B) $\frac{\pi}{4}$
(C) $\frac{\pi}{3}$
(D) $\frac{\pi}{2}$
(b) Obtain the value of $\Delta = \begin{vmatrix} 1+x & 1 & 1 \\ 1 & 1+y & 1 \\ 1 & 1 & 1+z \end{vmatrix}$ in terms of $x$, $y$ and $z$.
Further, if $\Delta = 0$ and $x$, $y$, $z$ are non-zero real numbers, prove that $x^{-1} + y^{-1} + z^{-1} = -1$.
If $A = \begin{bmatrix} \frac{1}{2}\cos x & -\sin x \\ \sin x & \frac{1}{2}\cos x \end{bmatrix}$ and $A + A' = I$, then value of $x \in \left[\frac{-\pi}{2}, \frac{\pi}{2}\right]$ is
(A) $\frac{\pi}{2}$
(B) $\frac{\pi}{3}$
(C) $0$
(D) $\frac{-\pi}{2}$
If A and B are square matrices of same order, then which of the following statements is/are always true ?
(i) $(\text{A} + \text{B}) (\text{A} - \text{B}) = \text{A}^2 - \text{B}^2$
(ii) $\text{AB} = \text{BA}$
(iii) $(\text{A} + \text{B})^2 = \text{A}^2 + \text{AB} + \text{BA} + \text{B}^2$
(iv) $\...
If $A = \begin{bmatrix} \tan x & \cot x \\ -\cot x & \tan x \end{bmatrix}$ and $A + A' = 2I$, then value of $x \in \left[0, \frac{\pi}{2}\right]$ is
(A) $0$
(B) $\frac{\pi}{3}$
(C) $\frac{\pi}{4}$
(D) $\frac{\pi}{2}$
(a) If $P = \begin{bmatrix} 1 & -1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2 \end{bmatrix}$ and $Q = \begin{bmatrix} 2 & 2 & -4 \\ -4 & 2 & -4 \\ 2 & -1 & 5 \end{bmatrix}$, find $( ext{QP})$ and hence solve the following system of equations using matrices :
$x - y = 3$, $2x + 3y + 4z = 17$, $y + 2z = 7$
OR
...
The number of all possible matrices of order $2 \times 3$ with each entry $1$ or $2$ is :
(A) $16$
(B) $81$
(C) $64$
(D) $32$
If $A = \begin{bmatrix} \cos 2\alpha & -\sin 2\alpha \\ \sin 2\alpha & \cos 2\alpha \end{bmatrix}$, then $A + A' = I$, if $\alpha$ is equal to :
(A) $\frac{\pi}{3}$
(B) $\frac{\pi}{6}$
(C) $\frac{\pi}{2}$
(D) $\frac{3\pi}{2}$
21. (a) For the matrix $A = \begin{bmatrix} 3 & 1 \\ 2 & 1 \end{bmatrix}$, find the numbers a and b so that $A^2 + aA + bI = O$.
OR
(b) Find the equation of line joining the points $(-1, -1)$ and $(1, 3)$, using determinants.
(a) If $A = \begin{bmatrix} 1 & -1 & 1 \\ 2 & 1 & -3 \\ 1 & 1 & 1 \end{bmatrix}$, find $A^{-1}$. Using $A^{-1}$ solve the system of equations :
$x - y + z = 4$
$2x + y - 3z = 0$
$x + y + z = 2$
OR
(b) If $A = \begin{bmatrix} 3 & 2 \\ 7 & 5 \end{bmatrix}$, $B = \begin{bmatrix} 6 & 7 \\ 8 & 9 \end{bmatrix}$, then show that $(AB)^{-1} = B^{-1}A^{-1}$.
If $\begin{bmatrix} 4 \\ 1 \\ 3 \end{bmatrix} \text{A} = \begin{bmatrix} -4 & 8 & 4 \\ -1 & 2 & 1 \\ -3 & 6 & 3 \end{bmatrix}$, then order of A must be :
(A) $3 \times 1$
(B) $1 \times 3$
(C) $1 \times 1$
(D) $3 \times 3$
If a square matrix A is such that $\text{A}^2 = \text{A}$ and $(\text{I} - \text{A})^3 = \text{xA} + \text{I}$, then value of x must be :
(A) $7$
(B) $5$
(C) $-7$
(D) $-1$
If $\text{B}(\text{adj } \text{B}) = \begin{bmatrix} \frac{1}{3} & 0 & 0 \\ 0 & \frac{1}{3} & 0 \\ 0 & 0 & \frac{1}{3} \end{bmatrix}$, then the value of $\text{det } (\text{B}^{-1}) =$
(A) $\frac{1}{3}$
(B) $\frac{1}{9}$
(C) $3$
(D) $9$
If A and B are skew-symmetric matrices of same order, then $\text{AB}' + \text{BA}'$ is a/an :
(A) symmetric matrix
(B) skew-symmetric matrix
(C) null matrix
(D) identity matrix
If a matrix B is such that $B\begin{bmatrix} 1 & 2 & 3 \end{bmatrix} = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 1 \\ 2 & 0 & 1 \end{bmatrix}$, then the order of matrix B is :
(A) $1 \times 3$
(B) $3 \times 1$
(C) $3 \times 3$
(D) $1 \times 1$
If a square matrix A is such that $A^2 = A$ and $(I - A)^3 = \text{xA} + I$, then value of x must be :
(A) $7$
(B) $5$
(C) $-7$
(D) $-1$
If $A(\text{adj } A) = \begin{bmatrix} 2026 & 0 & 0 \\ 0 & 2026 & 0 \\ 0 & 0 & 2026 \end{bmatrix}$, then the value of $|\text{adj } A|$ is equal to :
(A) $2026$
(B) $(2026)^{-1}$
(C) $(2026)^{-2}$
(D) $(2026)^2$
(a) On the inauguration day of a new showroom, a lucky draw was organized and some vouchers of ₹ 1,000 and ₹ 500 were given to the lucky draw winners.
A total of 60 vouchers were given on the day. The number of ₹ 1,000 vouchers added to 3 times the number of ₹ 500 vouchers, gives 100. Express the g...
If a matrix $X$ is such that $\begin{bmatrix} 2 & 1 \end{bmatrix} X = \begin{bmatrix} 3 & 4 & 5 \end{bmatrix}$, then the order of matrix $X$ is :
(A) $3 \times 1$
(B) $2 \times 3$
(C) $1 \times 2$
(D) $1 \times 3$
If a square matrix $A$ is such that $A^2 = A$ and $(I - A)^3 = xA + I$, then value of $x$ must be :
(A) $7$
(B) $5$
(C) $-7$
(D) $-1$
If $A(\text{adj } A) = \begin{bmatrix} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{bmatrix}$, then the value of $|2A|$ is :
(A) $6$
(B) $54$
(C) $12$
(D) $24$
OR Given that $P = \begin{bmatrix} 2 & -1 \\ 3 & 4 \end{bmatrix}$, $Q = \begin{bmatrix} 5 & 2 \\ 7 & 4 \end{bmatrix}$ and $R = \begin{bmatrix} 2 & 5 \\ 3 & 8 \end{bmatrix}$, find a matrix $S$ such that $PQ - RS$ is a zero matrix.
If $A = [a_{ij}]$ is a $2 \times 2$ matrix whose elements are given by $a_{ij} = \frac{|i - 3j|}{2}$, then $A'$ is :
(A) $\begin{bmatrix} 1 & \frac{5}{2} \\ \frac{1}{2} & 2 \end{bmatrix}$
(B) $\begin{bmatrix} 1 & \frac{1}{2} \\ \frac{5}{2} & 2 \end{bmatrix}$
(C) $\begin{bmatrix} 2 & \frac{5}{2} \\ \...
If $A = \begin{bmatrix} 3 & 2 & 0 \\ 1 & 4 & 0 \\ 0 & 0 & 5 \end{bmatrix}$, then compute $A^2 - 7A + 10 I$.
Let $A = [a_{ij}]$ be a $2 \times 2$ matrix whose elements are given by $a_{ij} = \frac{(2i - j)^2}{3}$. Then $A'$ is :
(A) $\begin{bmatrix} \frac{1}{3} & 3 \\ 0 & \frac{4}{3} \end{bmatrix}$
(B) $\begin{bmatrix} \frac{1}{3} & 0 \\ 3 & \frac{4}{3} \end{bmatrix}$
(C) $\begin{bmatrix} \frac{4}{3} & 3 \...
If $A = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix}$, then compute $A^2 - 4A - 5I$.
Let $A = [a_{ij}]$ be a $2 \times 2$ matrix whose elements are given by $a_{ij} = \frac{(i + 2j)^2}{3}$. Then $A'$ is given by: (A) $\begin{bmatrix} 12 & \frac{25}{3} \\ \frac{16}{3} & 3 \end{bmatrix}$ (B) $\begin{bmatrix} 12 & \frac{16}{3} \\ \frac{25}{3} & 3 \end{bmatrix}$ (C) $\begin{bmatrix} 3 &...
If A = $\begin{bmatrix} 2 & 0 & 1 \\ 2 & 1 & 3 \\ 1 & -1 & 0 \end{bmatrix}$, then compute $A^2 - 5A + 4 I$.
A carpenter needs to design a wooden box in the shape of a cuboid such that the sum of its length and breadth is $3 \text{ cm}$ more than its height. Twice of its length, thrice of its breadth and its height add up to $10 \text{ cm}$. Its breadth added to $7$ times its height is $1 \text{ cm}$ less ...
If matrix A = $\begin{bmatrix} -p & q \\ r & p \end{bmatrix}$ is such that $A^2 = I$, then : (A) $1 + p^2 + qr = 0$ (B) $1 - p^2 - qr = 0$ (C) $1 - p^2 + qr = 0$ (D) $1 + p^2 - qr = 0$
If A is a square matrix such that $A^2 = A$, then $(A - I)^3 - A$ is equal to : (A) I (B) $-I$ (C) A (D) $A^2$
A man goes to buy fruits from the market. The shopkeeper informs him that 4 apples, 3 oranges and 2 bananas cost ₹ 60; 2 apples, 4 oranges and 6 bananas cost ₹ 90; whereas 6 apples, 2 oranges and 3 bananas cost ₹ 70. Using matrix method, find the cost of one fruit of each kind.
If A = $\begin{bmatrix} \sin \theta & \cos \theta \\ -\cos \theta & \sin \theta \end{bmatrix}$ and A + A' = I, then $\theta$ is equal to :
(A) 0
(B) $\frac{\pi}{6}$
(C) $\frac{\pi}{3}$
(D) $\pi$
If A is a symmetric matrix, then for any matrix B of order same as A, BAB' is a/an :
(A) Skew symmetric matrix
(B) Identity matrix
(C) Symmetric matrix
(D) Null matrix
Find cost (per kg) of each fertilizer A, B and C that the farmer needs to buy, such that 1 kg each of fertilizer A and C added to 2 kg of B costs him ₹ 400. Also, cost of each kg of fertilizer B and C added together is equal to cost of 1 kg of fertilizer A. However, cost of 3 kg of fertilizer B adde...
For any square matrix $A$ with real entries, if $A + A'$ is a symmetric matrix then :
(A) $(A - A')$ cannot be a skew symmetric matrix
(B) $(A - A')$ is a skew symmetric matrix
(C) $A$ is always a symmetric matrix
(D) $A$ is always a skew symmetric matrix
A matrix $B = [b_{ij}]_{m \times m}$ is said to be a diagonal matrix, if :
(A) $b_{ij} = 0$, when $i = j$
(B) $b_{ij} = 1$, when $i = j$
(C) $b_{ij} = 1$, when $i \neq j$
(D) $b_{ij} = 0$, when $i \neq j$
Three students A, B and C go to a book-store to buy art books, story books and puzzle solving books. A buys one of each type of book for a total of ₹ 21. B buys 4 art books, 3 story books and 2 puzzle solving books for ₹ 60. C buys 6 art books, 2 story books and 3 puzzle solving books and pays ₹ 10 ...
(a) For matrices $A = \begin{bmatrix} 2 & 3 \\ 1 & -4 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & -2 \\ -1 & 3 \end{bmatrix}$, verify that $(\text{AB})^{-1} = \text{B}^{-1}\text{A}^{-1}$.
OR
(b) Using the matrix method, solve the system of equations :
$5x - 7y = 2$, $7x - 5y = 3$
(a) Find the adjoint of the matrix A given by,\n$A = \begin{bmatrix} 1 & 4 & 5 \\ 3 & 2 & 6 \\ 0 & 1 & 0 \end{bmatrix}$ and verify that\n$A(\text{adj } A) = |A| I = (\text{adj } A) A$.\nOR
(b) Show that the matrix $A = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix}$ satisfies the equation $A^2 - 4A - 5I_3 = \text{O}$ and hence find $A^{-1}$.
Students of a class were given a mathematical aptitude test. Their marks were found to be normally distributed with mean $60$ and standard deviation $5$. What percent of students scored :
(i) more than $60$ marks ?
(ii) less than $56$ marks ?
(iii) below $45$ and $65$ marks ?
$\text{Given} : \begin{...
If $\begin{vmatrix} 3 & x & -4 \\ 5 & -6 & 0 \\ -7 & 8 & 1 \end{vmatrix} = 0$, then the value of $x$ is :
(A) $2$
(B) $-2$
(C) $-\frac{26}{5}$
(D) $\frac{26}{5}$
If $A = \begin{bmatrix} 5 & 4 & 1 \end{bmatrix} \begin{bmatrix} 1 & 2 & 3 \\ 3 & 0 & 1 \\ 1 & 4 & 5 \end{bmatrix} \begin{bmatrix} 2 \\ 0 \\ 1 \end{bmatrix}$, then the order of A is :
(A) $1 \times 3$
(B) $1 \times 2$
(C) $3 \times 1$
(D) $1 \times 1$
The matrix $B = \begin{bmatrix} 0 & 0 & 7 \\ 0 & 7 & 0 \\ -7 & 0 & 0 \end{bmatrix}$ is a :
(A) skew symmetric matrix
(B) square matrix
(C) identity matrix
(D) diagonal matrix
Assertion ($A$) : If $\text{A}$ is a skew symmetric matrix of order $3$, then $|\text{A}| = 0$.
Reason ($R$) : If $\text{A}$ is a square matrix of order $2$, then $|\text{A}| = |\text{A}'|$.
(a) Find the product $\begin{bmatrix} 2 & 1 \\ 3 & 2 \\ -1 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 1 \\ -1 & 2 & 1 \end{bmatrix}$.
OR
(b) If $\text{A}' = \begin{bmatrix} 2 & 4 \\ 3 & 6 \\ 1 & 5 \end{bmatrix}$ and $\text{B} = \begin{bmatrix} 1 & 1 & 2 \\ -1 & 3 & 2 \end{bmatrix}$, then find $\tex...
Two motorcycle dealers, P and Q deal in two types of motorcycles, say A and B. The sales figures for the years 2022 and 2023 showed that dealer P sold 160 motorcycles of type A and 70 of type B in 2022 and 300 motorcycles of type A and 150 of type B in 2023. Dealer Q sold 100 motorcycles of type A a...
If $A$ is a square matrix, then which of the following is not true ?
(A) $A + A'$ is a symmetric matrix
(B) $AA'$ and $A'A$ are symmetric matrices
(C) $A + A'$ is a skew symmetric matrix
(D) $A - A'$ is a skew symmetric matrix
Solve the matrix equation for $x$ and $y$.
$$\begin{bmatrix} x^2 \\ y^2 \end{bmatrix} - 3 \begin{bmatrix} x \\ 3y \end{bmatrix} = \begin{bmatrix} -2 \\ -20 \end{bmatrix}$$
Solve the given matrix equation for x, given x > 0 :
$x \begin{bmatrix} 2x & 2 \\ 3 & x \end{bmatrix} + 2 \begin{bmatrix} 8 & 5x \\ 4 & 0 \end{bmatrix} = 2 \begin{bmatrix} x^2+8 & 24 \\ 10 & 8 \end{bmatrix}$
Find the value of $x$ if $A = B$, where $A = \begin{bmatrix} 6 & 3x \\ 8 & -6 \end{bmatrix}$ and $B = \begin{bmatrix} 6 & x^2 + 2 \\ 8 & x^2 - 5x \end{bmatrix}$.
If for three matrices $A = [a_{ij}]_{m \times 4}$, $B = [b_{ij}]_{n \times 3}$ and $C = [c_{ij}]_{p \times q}$ products $AB$ and $AC$ both are defined and are square matrices of same order, then value of $m, n, p$ and $q$ are:\n(A) $m = q = 3 \text{ and } n = p = 4$\n(B) $m = 2, q = 3 \text{ and } n...
If the matrix $A = \begin{bmatrix} 0 & r & -2 \\ 3 & p & t \\ q & -4 & 0 \end{bmatrix}$ is skew-symmetric, then value of $\frac{q+t}{p+r}$ is....\n(A) $-2$\n(B) $0$\n(C) $1$\n(D) $2$
If $A$ is a square matrix of order $4$ and $|\text{adj } A| = 27$, then $A (\text{adj } A)$ is equal to\n(A) $3$\n(B) $9$\n(C) $3I$\n(D) $9I$
The inverse of the matrix $\begin{bmatrix} 3 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 5 \end{bmatrix}$ is...\n(A) $\begin{bmatrix} 0 & 0 & 3 \\ 0 & 2 & 0 \\ 5 & 0 & 0 \end{bmatrix}$\n(B) $\begin{bmatrix} \frac{1}{3} & 0 & 0 \\ 0 & \frac{1}{2} & 0 \\ 0 & 0 & \frac{1}{5} \end{bmatrix}$\n(C) $\begin{bmatrix} -\...
Value of the determinant $\begin{vmatrix} \cos 67^{\circ} & \sin 67^{\circ} \\ \sin 23^{\circ} & \cos 23^{\circ} \end{vmatrix}$ is\n(A) $0$\n(B) $\frac{1}{2}$\n(C) $\frac{\sqrt{3}}{2}$\n(D) $1$
For two matrices $A = \begin{bmatrix} 3 & -6 & -1 \\ 2 & -5 & -1 \\ -2 & 4 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & -2 & -1 \\ 0 & -1 & -1 \\ 2 & 0 & 3 \end{bmatrix}$, find the product $AB$ and hence solve the system of equations:\n$3x - 6y - z = 3$\n$2x - 5y - z + 2 = 0$\n$-2x + 4y + z = 5$
If $\text{P} = \begin{bmatrix} 1 & 0 \\ 2 & 1 \end{bmatrix}$, $\text{Q} = \begin{bmatrix} x & 0 \\ 1 & 1 \end{bmatrix}$ and $\text{P} = \text{Q}^2$, then $x$ equals :
(A) $\pm 1$
(B) $-1$
(C) $1$
(D) $2$
If $\text{A}$ is an invertible matrix, then which of the following is not true ?
(A) $|\text{A}^{-1}| = |\text{A}|^{-1}$
(B) $(\text{A}^2)^{-1} = (\text{A}^{-1})^2$
(C) $(\text{A}')^{-1} = (\text{A}^{-1})'$
(D) $|\text{A}| \neq 0$
Questions number 19 and 20 are Assertion and Reason based questions. Two statements are given, one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the codes (A), (B), (C) and (D) as given below.
(A) Both Assertion (A) and Reason (R) are true and Reason (R) is...
(a) Prove that $\begin{vmatrix} x+y & x & x \\ 5x+4y & 4x & 2x \\ 10x+8y & 8x & 3x \end{vmatrix} = x^3$
OR
(b) Prove that $\begin{vmatrix} y+z & z & y \\ z & z+x & x \\ y & x & x+y \end{vmatrix} = 4xyz$
Find the inverse (if it exists) of the matrix $A = \begin{bmatrix} 1 & 2 & -2 \\ -1 & 3 & 0 \\ 0 & -2 & 1 \end{bmatrix}$.
If $A = \begin{bmatrix} x & 3 \\ 3 & x \end{bmatrix}$ and $|A^3| = 343$, then $x$ is :
(A) $\pm 7$
(B) $\pm 4$
(C) $\pm 3$
(D) $\pm 5$
If $A$ and $B$ are square matrices both of order $3$, such that $|A| = -3$ and $|B| = 2$, then $|2AB|$ is equal to :
(A) $48$
(B) $-48$
(C) $-24$
(D) $-12$
For a skew-symmetric matrix $A = \begin{bmatrix} 0 & 7 & -4 \\ r & p & 5 \\ q & -5 & 0 \end{bmatrix}$, the value of $p + q - r$ is :
(A) $11$
(B) $-3$
(C) $-11$
(D) $3$
Assertion (A) : If $A$ is a skew-symmetric matrix of order $3$, then $|A| = 0$.
Reason (R) : If $A$ is a square matrix of order $3$, then $|A| = |A'|$.
To promote the making of toilets in villages, an NGO hired an agency for generating awareness for the cause through house calls, letters and announcements through speakers.
The cost per mode of communication is given below :
| Cost per visit / communication | House calls | Letters | Announcements |...
If the matrix $A = [a_{ij}]_{2 \times 2}$ is such that $a_{ij} = \begin{cases} 1, & \text{if } i \neq j \\ 0, & \text{if } i = j \end{cases}$, then $A + A^2$ is equal to :
(A) $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$
(B) $\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$
(C) $\begin{bmatrix} 1 & 1...
The cofactor of the element $a_{32}$ in the determinant $\Delta = \begin{vmatrix} 1 & 2 & 3 \\ 2 & 3 & 1 \\ 3 & 1 & 2 \end{vmatrix}$ is :
(A) $\pm 5$
(B) $-5$
(C) $5$
(D) $0$
If A is an identity matrix of order n, then $A(\text{Adj } A)$ is a/an :
(A) identity matrix
(B) row matrix
(C) zero matrix
(D) skew symmetric matrix
If $A = \begin{bmatrix} 3 & 2 & 1 \\ 4 & -1 & 2 \\ 7 & 3 & -3 \end{bmatrix}$, find $A^{-1}$. Using $A^{-1}$, solve the given system of equations $3x + 4y + 7z = 14$; $2x - y + 3z = 4$; $x + 2y - 3z = 0$.
If $\begin{bmatrix} 2\text{x} & 3 \end{bmatrix} \begin{bmatrix} \text{x} \\ -8 \end{bmatrix} = 0$, then the value of $\text{x}$ is :
(A) zero
(B) $3$
(C) $2\sqrt{3}$
(D) $\pm 2\sqrt{3}$
If A is an identity matrix of order n, then A (Adj A) is a/an :
(A) identity matrix
(B) row matrix
(C) zero matrix
(D) skew symmetric matrix
If $A = \begin{bmatrix} 2 & -3 & 5 \\ 3 & 2 & -4 \\ 1 & 1 & -2 \end{bmatrix}$, find $A^{-1}$. Using $A^{-1}$, solve the given system of equations :
$2x - 3y + 5z = 11$
$3x + 2y - 4z = -5$
$x + y - 2z = -3$.
The matrix $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4 \end{bmatrix}$ is :
(A) an identity matrix
(B) a scalar matrix
(C) a symmetric matrix
(D) a skew-symmetric matrix
If $\begin{bmatrix} 3x & 5 \end{bmatrix} \begin{bmatrix} x \\ -15 \end{bmatrix} = 0$, then the value of $x$ is :
(A) zero
(B) $5$
(C) $2\sqrt{5}$
(D) $\pm 5$
If $A = \begin{bmatrix} 1 & 2 \\ 4 & 2 \end{bmatrix}$ and $|3A| = k |A|$, then the value of $k$ is :
(A) $3$
(B) $9$
(C) $6$
(D) $27$
If $A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}$, then the value of $|\text{adj } A|$ is :
(A) $5$
(B) $25$
(C) $-5$
(D) $-25$
If $A = \begin{bmatrix} 2 & 2 & -4 \\ -4 & 2 & -4 \\ 2 & -1 & 5 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & -1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2 \end{bmatrix}$, find BA and use this to solve the system of equations :
$y + 2z = 8$,
$x - y = -1$,
$2x + 3y + 4z = 20$
If $\begin{bmatrix} x+y & 2 \\ 5 & xy \end{bmatrix} = \begin{bmatrix} 6 & 2 \\ 5 & 8 \end{bmatrix}$, then the value of $\left$\frac{24}{\text{x}} + \frac{24}{\text{y}}\right$$ is :
(A) $7$
(B) $6$
(C) $8$
(D) $18$
$\begin{vmatrix} x + 1 & x - 1 \\ x^2 + x + 1 & x^2 - x + 1 \end{vmatrix}$ is equal to :
(A) $2x^3$
(B) $2$
(C) $0$
(D) $2x^3 - 2$
If $\text{A}$ and $\text{B}$ are two non-zero square matrices of same order such that $(\text{A} + \text{B})^2 = \text{A}^2 + \text{B}^2$, then :
(A) $\text{AB} = \text{O}$
(B) $\text{AB} = -\text{BA}$
(C) $\text{BA} = \text{O}$
(D) $\text{AB} = \text{BA}$
(a) If $A = \begin{bmatrix} 1 & -2 & 0 \\ 2 & -1 & -1 \\ 0 & -2 & 1 \end{bmatrix}$, find $A^{-1}$ and use it to solve the following system of equations : $x - 2y = 10, 2x - y - z = 8, -2y + z = 7$ \
OR \
(b) If $A = \begin{bmatrix} -1 & a & 2 \\ 1 & 2 & x \\ 3 & 1 & 1 \end{bmatrix}$ and $A^{-1} = ...
If A is a square matrix of order 2 such that $\text{det } (\text{A}) = 4$, then $\text{det } (4 \text{ adj A})$ is equal to :
(A) $16$
(B) $64$
(C) $256$
(D) $512$
Let $A = \begin{bmatrix} 1 & -2 & -1 \\ 0 & 4 & -1 \\ -3 & 2 & 1 \end{bmatrix}$, $B = \begin{bmatrix} -2 \\ -5 \\ -7 \end{bmatrix}$, $C = \begin{bmatrix} 9 & 8 & 7 \end{bmatrix}$, which of the following is defined ?
(A) Only AB
(B) Only AC
(C) Only BA
(D) All AB, AC and BA
If $A = \begin{bmatrix} 7 & 0 & x \\ 0 & 7 & 0 \\ 0 & 0 & y \end{bmatrix}$ is a scalar matrix, then $y^x$ is equal to
(A) $0$
(B) $1$
(C) $7$
(D) $\pm 7$
Let $A$ be a matrix of order $m \times n$ and $B$ is a matrix such that $A^{\text{T}}B$ and $BA^{\text{T}}$ are defined. Then, the order of $B$ is :
(A) $m \times m$
(B) $n \times n$
(C) $m \times n$
(D) $n \times m$
If $\begin{bmatrix} x + y & 3y \\ 3x & x + 3 \end{bmatrix} = \begin{bmatrix} 9 & 4x + y \\ x + 6 & y \end{bmatrix}$, then $(x - y) = ?$
(A) $-7$
(B) $-3$
(C) $3$
(D) $7$
Let $A = \begin{bmatrix} 1 & -2 & -1 \\ 0 & 4 & -1 \\ -3 & 2 & 1 \end{bmatrix}$, $B = \begin{bmatrix} -2 \\ -5 \\ -7 \end{bmatrix}$, $C = [9 \text{ } 8 \text{ } 7]$, which of the following is defined?
(A) Only AB
(B) Only AC
(C) Only BA
(D) All AB, AC and BA
If $A = \begin{bmatrix} 1 & 12 & 4y \\ 6x & 5 & 2x \\ 8x & 4 & 6 \end{bmatrix}$ is a symmetric matrix, then $(2x + y)$ is
(A) $-8$
(B) $0$
(C) $6$
(D) $8$
Assertion (A) : $A = \text{diag} \, [3 \;\; 5 \;\; 2]$ is a scalar matrix of order $3 \times 3$.
Reason (R) : If a diagonal matrix has all non-zero elements equal, it is known as a scalar matrix.
Let $P$ be a skew-symmetric matrix of order $3$. If $\det(P) = \alpha$, then $(2025)^\alpha$ is
(A) $0$
(B) $1$
(C) $2025$
(D) $(2025)^3$
Assertion (A) : A = diag [ 3 5 2] is a scalar matrix of order $3 \times 3$.
Reason (R) : If a diagonal matrix has all non-zero elements equal, it is known as a scalar matrix.
If $A = \begin{bmatrix} x & 0 & m \\ y & z & 0 \\ 0 & 0 & 6 \end{bmatrix} = 6I$, where $I$ is a unit matrix, then $x + y + z + m$ is equal to
(A) $18$
(B) $12$
(C) $6$
(D) $2$
If $B = \begin{bmatrix} 23 & 41 & 57 \end{bmatrix} \begin{bmatrix} 31 & 42 \\ 53 & 64 \\ 75 & 86 \end{bmatrix}$, then the order of $B$ is:
(A) $3 \times 2$
(B) $2 \times 2$
(C) $1 \times 3$
(D) $1 \times 2$
If $A$ and $B$ are square matrices of the same order, then $(A - B)^2 = ?$
(A) $A^2 - 2AB + B^2$
(B) $A^2 - AB - BA + B^2$
(C) $A^2 - 2BA + B^2$
(D) $A^2 - AB + BA + B^2$
If $\begin{vmatrix} 5 & 3 & -1 \\ -7 & x & 2 \\ 9 & 6 & -2 \end{vmatrix} = 0$, then the value of $x$ is
(A) $0$
(B) $9$
(C) $-6$
(D) $6$
If $A^{-1} = \begin{bmatrix} 7 & 2 \\ 8 & 2 \end{bmatrix}$, then matrix A is
(A) $\begin{bmatrix} 2 & -2 \\ -8 & 7 \end{bmatrix}$
(B) $\begin{bmatrix} -7 & 8 \\ 2 & -2 \end{bmatrix}$
(C) $\begin{bmatrix} -1 & 1 \\ 4 & -\frac{7}{2} \end{bmatrix}$
(D) $\begin{bmatrix} 1 & -1 \\ -4 & \frac{7}{2} \e...
If A = $\begin{bmatrix} 2 & -3 & 5 \\ 3 & 2 & -4 \\ 1 & 1 & -2 \end{bmatrix}$, then find $A^{-1}$. Using $A^{-1}$, solve the system of equations :
$2x - 3y + 5z = 11$
$3x + 2y - 4z = -5$
$x + y - 2z = -3$
If A and B are square matrices of same order such that $\text{AB} = \text{A}$ and $\text{BA} = \text{B}$, then $\text{A}^2 + \text{B}^2$ is equal to :
(A) $\text{A} + \text{B}$
(B) $\text{BA}$
(C) $2(\text{A} + \text{B})$
(D) $2\text{BA}$
If $M$ and $N$ are square matrices of order $3$ such that $\text{det } (M) = m$ and $MN = mI$, then $\text{det } (N)$ is equal to: (A) $-1$ (B) $1$ (C) $-m^2$ (D) $m^2$
The matrix [ 0 1 -2 ; -1 0 -7 ; 2 7 0 ] is a : (A) diagonal matrix (B) symmetric matrix (C) skew symmetric matrix (D) scalar matrix
Let A and B be two square matrices of order 3 such that $\text{det (A) = 3}$ and $\text{det (B) = - 4}$. Find the value of $\text{det (- 6AB)}$.
If $\begin{vmatrix} -1 & 2 & 4 \\ 1 & x & 1 \\ 0 & 3 & 3x \end{vmatrix} = -57$, the product of the possible values of x is : (A) $-24$ (B) $-16$ (C) 16 (D) 24
Using matrices and determinants, find the value(s) of k for which the pair of equations $5x - ky = 2$; $7x - 5y = 3$ has a unique solution.
If $A = \begin{bmatrix} 2 & -2 \\ -2 & 2 \end{bmatrix}$ and $A^2 = kA$, then find the value of k.
If A = $\begin{bmatrix} 1 & 2 & 3 \\ -4 & 3 & 7 \end{bmatrix}$ and B = $\begin{bmatrix} 4 & 3 \\ -1 & 2 \\ 0 & 5 \end{bmatrix}$, then the correct statement is :
(A) Only AB is defined.
(B) Only BA is defined.
(C) AB and BA, both are defined.
(D) AB and BA, both are not defined.
If $\begin{bmatrix} 4+x & x-1 \\ -2 & 3 \end{bmatrix}$ is a singular matrix, then the value of x is :
(A) 0
(B) 1
(C) -2
(D) -4
A furniture workshop produces three types of furniture - chairs, tables and beds each day. On a particular day the total number of furniture pieces produced is 45. It was also found that production of beds exceeds that of chairs by 8, while the total production of beds and chairs together is twice t...
If A and B are two square matrices of the same order, then (A + B) (A - B) is equal to :
(A) $A^2 - AB + BA - B^2$
(B) $A^2 + AB - BA - B^2$
(C) $A^2 - AB - BA - B^2$
(D) $A^2 - B^2 + AB + BA$
If a matrix A is both symmetric and skew-symmetric, then A is a :
(A) diagonal matrix
(B) zero matrix
(C) non-singular matrix
(D) scalar matrix
An amount of ₹ 10,000 is put into three investments at the rate of 10%, 12% and 15% per annum. The combined annual income of all three investments is ₹ 1,310, however the combined annual income of the first and the second investments is ₹ 190 short of the income from the third. Use matrix method and...
If $\text{A} = [\text{a}_{ij}]$ is a $3 \times 3$ diagonal matrix such that $\text{a}_{11} = 1, \text{a}_{22} = 5$ and $\text{a}_{33} = -2$, then $|\text{A}|$ is :
(A) 0
(B) $-10
(C) 10
(D) 1
If $\text{A} = \text{kB}$, where $\text{A}$ and $\text{B}$ are two square matrices of order $\text{n}$ and $\text{k}$ is a scalar, then :
(A) $|\text{A}| = \text{k}|\text{B}|$
(B) $|\text{A}| = \text{k}^{\text{n}}|\text{B}|$
(C) $|\text{A}| = \text{k} + |\text{B}|$
(D) $|\text{A}| = |\text{B}|^{\tex...
Let A and B be two matrices of suitable orders. Then, which of the following is not correct ?
(A) (A')' = A
(B) (kA)' = kA', k is a scalar
(C) (A' + B')' = A + B
(D) (AB)' = A'B'
Three students run on a racing track such that their speeds add up to $6\text{ km/h}$. However, double the speed of the third runner added to the speed of the first results in $7\text{ km/h}$. If thrice the speed of the first runner is added to the original speeds of the other two, the result is $12...
If A = $\begin{bmatrix} -1 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \end{bmatrix}$, then A is a/an :
(A) scalar matrix
(B) identity matrix
(C) symmetric matrix
(D) skew-symmetric matrix
Let $A = \begin{bmatrix} 1 \\ 4 \\ -2 \end{bmatrix}$ and $C = \begin{bmatrix} 3 & 4 & 2 \\ 12 & 16 & 8 \\ -6 & -8 & -4 \end{bmatrix}$ be two matrices. Then, find the matrix B if $\text{AB} = \text{C}$.
If A = $\begin{bmatrix} 0 & -3 & 8 \\ 3 & 0 & 5 \\ -8 & -5 & 0 \end{bmatrix}$, then A is a : (A) null matrix (B) symmetric matrix (C) skew-symmetric matrix (D) diagonal matrix
If $A = \begin{bmatrix} 1 & -1 & 0 \end{bmatrix}$, $B = \begin{bmatrix} 2 & 0 & 1 \\ -1 & 3 & 4 \\ 0 & 5 & 1 \end{bmatrix}$ and $C = \begin{bmatrix} 2 \\ 3 \\ 4 \end{bmatrix}$, are three matrices, then find $\text{ABC}$.
Three students, Neha, Rani and Sam go to a market to purchase stationery items. Neha buys $4$ pens, $3$ notepads and $2$ erasers and pays $\text{₹ } 60$. Rani buys $2$ pens, $4$ notepads and $6$ erasers for $\text{₹ } 90$. Sam pays $\text{₹ } 70$ for $6$ pens, $2$ notepads and $3$ erasers.
Based upo...
If $A = \begin{bmatrix} 0 & 0 & -5 \\ 0 & 3 & 0 \\ 4\cdot3 & 0 & 0 \end{bmatrix}$, then $A$ is a :
(A) skew-symmetric matrix
(B) scalar matrix
(C) diagonal matrix
(D) square matrix
Find the value of $\text{x}$, if $[1 \quad \text{x} \quad 1] \begin{bmatrix} 1 & 3 & 2 \\ 2 & 5 & 1 \\ 15 & 3 & 2 \end{bmatrix} \begin{bmatrix} 1 \\ 2 \\ \text{x} \end{bmatrix} = O$.
What is the total number of possible matrices of order $3 \times 3$ with each entry as $\sqrt{2}$ or $\sqrt{3}$ ?
The matrix $A = \begin{bmatrix} \sqrt{3} & 0 & 0 \\ 0 & \sqrt{2} & 0 \\ 0 & 0 & \sqrt{5} \end{bmatrix}$ is a/an :
If $A$ and $B$ are two square matrices each of order $3$ with $|A| = 3$ and $|B| = 5$, then $|2AB|$ is :
Let $A$ be a square matrix of order $3$. If $|A| = 5$, then $|\text{adj } A|$ is :
If $\begin{bmatrix} 2x - 1 & 3x \\ 0 & y^2 - 1 \end{bmatrix} = \begin{bmatrix} x + 3 & 12 \\ 0 & 35 \end{bmatrix}$, then the value of $(x - y)$ is :
If $A = \begin{bmatrix} 2 & 3 \\ -1 & 2 \end{bmatrix}$, then show that $A^2 - 4A + 7I = 0$.
If $\begin{bmatrix} 2x - 1 & 3x \\ 0 & y^2 - 1 \end{bmatrix} = \begin{bmatrix} x + 3 & 12 \\ 0 & 35 \end{bmatrix}$, then the value of $(x - y)$ is :
(A) 2 or 10
(B) -2 or 10
(C) 2 or -10
(D) -2 or -10
If $\begin{bmatrix} 3 & -1 \\ 0 & 1 \\ 2 & -3 \end{bmatrix} A = \begin{bmatrix} 2 \\ -5 \\ -17 \end{bmatrix}$, then find matrix A.
26. (a) Let $2x + 5y - 1 = 0$ and $3x + 2y - 7 = 0$ represent the equations of two lines on which the ants are moving on the ground. Using matrix method, find a point common to the paths of the ants.\nOR\n(b) A shopkeeper sells $50$ Chemistry, $60$ Physics and $35$ Maths books on day I and sells $40...
Let A be a square matrix of order 3. If $|A| = 5$, then $|\text{adj } A|$ is : (A) 5 (B) 125 (C) 25 (D) $-5$
If $\begin{bmatrix} 2x - 1 & 3x \\ 0 & y^2 - 1 \end{bmatrix} = \begin{bmatrix} x + 3 & 12 \\ 0 & 35 \end{bmatrix}$, then the value of $(x - y)$ is :
(A) $2 \text{ or } 10$
(B) $-2 \text{ or } 10$
(C) $2 \text{ or } -10$
(D) $-2 \text{ or } -10$
If $A = \begin{bmatrix} 1 & 0 \\ -1 & 5 \end{bmatrix}$, then find the value of $K$ if $A^2 = 6A + KI_2$, where $I_2$ is an identity matrix.
(b) A shopkeeper sells 50 Chemistry, 60 Physics and 35 Maths books on day I and sells 40 Chemistry, 45 Physics and 50 Maths books on day II. If the selling price for each such subject book is ₹ 150 (Chemistry), ₹ 175 (Physics) and ₹ 180 (Maths), then find his total sale in two days, using matrix met...
If matrices $A$ and $B$ are of order $1 \times 3$ and $3 \times 1$ respectively, then the order of $A'B'$ is :
(A) $1 \times 1$
(B) $3 \times 1$
(C) $1 \times 3$
(D) $3 \times 3$
If $\begin{bmatrix} x+y & 2 \\ 5 & xy \end{bmatrix} = \begin{bmatrix} 6 & 2 \\ 5 & 8 \end{bmatrix}$, then the value of $\left$\frac{24}{x} + \frac{24}{y}\right$$ is :
(A) $7$
(B) $6$
(C) $8$
(D) $18$
If $\begin{vmatrix} x + 1 & x - 1 \\ x^2 + x + 1 & x^2 - x + 1 \end{vmatrix}$ is equal to :
(A) $2x^3$
(B) $2$
(C) $0$
(D) $2x^3 - 2$
(a) If A = $\begin{bmatrix} 1 & -2 & 0 \\ 2 & -1 & -1 \\ 0 & -2 & 1 \end{bmatrix}$, find $A^{-1}$ and use it to solve the following system of equations : x - 2y = 10, 2x - y - z = 8, - 2y + z = 7
OR
(b) If A = $\begin{bmatrix} -1 & \text{a} & 2 \\ 1 & 2 & \text{x} \\ 3 & 1 & 1 \end{bmatrix}$ and $A^...
If $A = \begin{bmatrix} 0 & 1 & c \\ -1 & a & -b \\ 2 & 3 & 0 \end{bmatrix}$ is a skew-symmetric matrix then the value of $a+b+c = $ (A) $1$ (B) $2$ (C) $3$ (D) $4$
The equation of the path traversed by the ball headed by the footballer is $y = ax^2 + bx + c$; (where $0 \leq x \leq 14$ and $a, b, c \in R$ and $a \neq 0$) with respect to a XY-coordinate system in the vertical plane. The ball passes through the points $(2, 15)$, $(4, 25)$ and $(14, 15)$. Determin...
If A = [a$_{ij}$] be a 3 × 3 matrix, where a$_{ij}$ = i − 3j, then which of the following is false ? (A) a$_{11}$ < 0 (B) a$_{12}$ + a$_{21}$ = − 6 (C) a$_{13}$ > a$_{31}$ (D) a$_{31}$ = 0
The number of all scalar matrices of order 3, with each entry $-1, 0$ or $1$, is :
(A) $1$
(B) $3$
(C) $2$
(D) $3^9$
If $A = \begin{bmatrix} 5 & 0 & 4 \\ 2 & 3 & 2 \\ 1 & 2 & 1 \end{bmatrix}$ and $B^{-1} = \begin{bmatrix} 1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4 \end{bmatrix}$, find $(AB)^{-1}$. Also, find $|(AB)^{-1}|$.
Given $A = \begin{bmatrix} 1 & 1 & 1 \\ 2 & 3 & 2 \\ 1 & 1 & 2 \end{bmatrix}$, find $A^{-1}$. Use it to solve the following system of equations: $x + y + z = 1$, $2x + 3y + 2z = 2$, $x + y + 2z = 4$
If for the matrix $A = \begin{bmatrix} \tan x & 1 \\ -1 & \tan x \end{bmatrix}$, $A + A' = 2\sqrt{3} I$, then the value of $x \in [0, \frac{\pi}{2}]$ is :
(A) 0
(B) $\frac{\pi}{4}$
(C) $\frac{\pi}{3}$
(D) $\frac{\pi}{6}$
If $A = [a_{ij}]$ be a $3 \times 3$ matrix, where $a_{ij} = i - 3j$, then which of the following is false ?
(A) $a_{11} < 0$ (B) $a_{12} + a_{21} = -6$
(C) $a_{13} > a_{31}$ (D) $a_{31} = 0$
If A = $\begin{bmatrix} \cos x & -\sin x \\ \sin x & \cos x \end{bmatrix}$, then the value of x, for which A is an identity matrix, is
(A) $\frac{\pi}{2}$
(B) $\pi$
(C) 0
(D) $\frac{3\pi}{2}$
If the matrix A = $\begin{bmatrix} 0 & 5 & -7 \\ a & 0 & 3 \\ b & -3 & 0 \end{bmatrix}$ is a skew-symmetric matrix, then the values of 'a' and 'b' are : (A) a = 5, b = 3 (B) a = 5, b = -7 (C) a = -5, b = -7 (D) a = -5, b = 7
If $\begin{vmatrix} x + 2 & x - 4 \\ x - 2 & x + 3 \end{vmatrix} = \begin{vmatrix} 6 & -2 \\ 1 & 3 \end{vmatrix}$, then the value of x is : (A) 1 (B) 2 (C) -2 (D) -1
If $\begin{bmatrix} 8 & 14 \\ 9 & 7 \end{bmatrix} = \begin{bmatrix} 1 & 2 \\ 3 & 1 \end{bmatrix} X$, then matrix X is : (A) $\begin{bmatrix} 3 & 7 \\ 2 & 0 \end{bmatrix}$ (B) $\begin{bmatrix} 2 & 0 \\ 7 & 3 \end{bmatrix}$ (C) $\begin{bmatrix} 2 & 0 \\ 3 & 7 \end{bmatrix}$ (D) $\begin{bmatrix} 2 & 0 ...
Using matrices, solve the following system of linear equations : $3x + 4y + 2z = 8 \; ; \; 2y - 3z = 3 \; ; \; x - 2y + 6z = - 2$
For any square matrix $A$, $(A - A')$ is always
(A) an identity matrix
(B) a null matrix
(C) a skew symmetric matrix
(D) a symmetric matrix
Case Study - 2
A scholarship is a sum of money provided to a student to help him or her pay for education. Some students are granted scholarships based on their academic achievements, while others are rewarded based on their financial needs.
Every year a school offers scholarships to girl children...
The value of $\begin{vmatrix} 8 & 2 & 7 \\ 12 & 3 & 5 \\ 16 & 4 & 3 \end{vmatrix}$ is :
(A) 0
(B) 2
(C) 7
(D) -2
If $A = \begin{bmatrix} 2 & 1 \\ -4 & -2 \end{bmatrix}$, then the value of $I - A + A^2 - A^3 + ...$ is : (A) $\begin{bmatrix} -1 & -1 \\ 4 & 3 \end{bmatrix}$ (B) $\begin{bmatrix} 3 & 1 \\ -4 & -1 \end{bmatrix}$ (C) $\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$ (D) $\begin{bmatrix} 1 & 0 \\ 0 & 1 \...
If $A = \begin{bmatrix} -2 & 0 & 0 \\ 1 & 2 & 3 \\ 5 & 1 & -1 \end{bmatrix}$, then the value of $| A(\text{adj. } A) |$ is : (A) $100 I$ (B) $10 I$ (C) $10$ (D) $1000$
Given that $[1 \quad x] \begin{bmatrix} 4 & 0 \\ -2 & 0 \end{bmatrix} = 0$, the value of $x$ is : (A) $-4$ (B) $-2$ (C) $2$ (D) $4$
If $A = \begin{bmatrix} 2 & 1 & -3 \\ 3 & 2 & 1 \\ 1 & 2 & -1 \end{bmatrix}$, find $A^{-1}$ and hence solve the following system of equations : $2x + y - 3z = 13$, $3x + 2y + z = 4$, $x + 2y - z = 8$
If $A = [a_{ij}] = \begin{bmatrix} 2 & -1 & 5 \\ 1 & 3 & 2 \\ 5 & 0 & 4 \end{bmatrix}$ and $c_{ij}$ is the cofactor of element $a_{ij}$, then the value of $a_{21} \cdot c_{11} + a_{22} \cdot c_{12} + a_{23} \cdot c_{13}$ is :
(A) $-57$
(B) 0
(C) 9
(D) 57
If A = \begin{bmatrix} 1 & 3 \\ 3 & 4 \end{bmatrix} and A^2 - kA - 5I = O, then the value of k is : (A) 3 (B) 5 (C) 7 (D) 9
Use the product of matrices $\begin{pmatrix} 1 & 2 & -3 \\ 3 & 2 & -2 \\ 2 & -1 & 1 \end{pmatrix} \begin{pmatrix} 0 & 1 & 2 \\ -7 & 7 & -7 \\ -7 & 5 & -4 \end{pmatrix}$ to solve the following system of equations :
$x + 2y - 3z = 6$
$3x + 2y - 2z = 3$
$2x - y + z = 2$
If $A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}$ and $A^2 + 7I = kA$, then the value of $k$ is :
(A) 1
(B) 2
(C) 5
(D) 7
Let $A = \begin{bmatrix} 1 & -1 & 2 \\ 0 & 2 & -3 \\ 3 & -2 & 4 \end{bmatrix}$ and $B = \frac{1}{3} \begin{bmatrix} -2 & 0 & 1 \\ 9 & 2 & -3 \\ 6 & 1 & \lambda \end{bmatrix}$. If $AB = I$, then the value of $\lambda$ is :
(A) $\frac{-9}{4}$
(B) -2
(C) $\frac{-3}{2}$
(D) 0
Find $A^{-1}$, if $A = \begin{bmatrix} 1 & 2 & 1 \\ 2 & 3 & -1 \\ 1 & 0 & 1 \end{bmatrix}$. Hence, solve the following system of equations :
$x + 2y + z = 5$
$2x + 3y = 1$
$x - y + z = 8$
(a) If $A = \begin{bmatrix} 1 & 2 & -3 \\ 2 & 0 & -3 \\ 1 & 2 & 0 \end{bmatrix}$, then find $A^{-1}$ and hence solve the following system of equations : $x + 2y - 3z = 1$ $2x - 3z = 2$ $x + 2y = 3$ OR (b) Find the product of the matrices $\begin{bmatrix} 1 & 2 & -3 \\ 2 & 3 & 2 \\ 3 & -3 & -4 \end{b...
OR (b) Find the product of the matrices $\begin{bmatrix} 1 & 2 & -3 \\ 2 & 3 & 2 \\ 3 & -3 & -4 \end{bmatrix} \begin{bmatrix} -6 & 17 & 13 \\ 14 & 5 & -8 \\ -15 & 9 & -1 \end{bmatrix}$ and hence solve the system of linear equations : $x + 2y - 3z = -4$, $2x + 3y + 2z = 2$, $3x - 3y - 4z = 11$.
If $A$ is a square matrix of order $3$ such that the value of $|\text{adj} \cdot A| = 8$, then the value of $|A^T|$ is :
(A) $\sqrt{2}$
(B) $-\sqrt{2}$
(C) $8$
(D) $2\sqrt{2}$
If A = $\begin{bmatrix} 4 & 1 \\ 3 & 2 \end{bmatrix}$ and I = $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, then $(A^2 - 6A)$ is equal to :
(A) 3I
(B) $-5I$
(C) 5I
(D) $-3I$
If A = $\begin{bmatrix} 2x & 0 \\ x & x \end{bmatrix}$ and $A^{-1} = \begin{bmatrix} 1 & 0 \\ -1 & 2 \end{bmatrix}$, then the value of x is :
(A) 1
(B) $\frac{1}{2}$
(C) $-\frac{1}{2}$
(D) 2
Assertion (A) : Minor of element $a_{13}$ in the matrix $\begin{bmatrix} 0 & 2 & 6 \\ 1 & 2 & -1 \\ 2 & 1 & 3 \end{bmatrix}$ is $\begin{vmatrix} 1 & 2 \\ 2 & 1 \end{vmatrix}$. Reason (R) : Minor of an element $a_{ij}$ of a matrix is the determinant obtained by deleting its $j^{th}$ row and $i^{th}$ ...
(a) Evaluate : $\begin{vmatrix} 1 & 1 & 1 \\ x & y & z \\ x^2 & y^2 & z^2 \end{vmatrix}$
OR
(b) Find the inverse of the matrix $\begin{bmatrix} 1 & 2 & -2 \\ -1 & 3 & 0 \\ 0 & -2 & 1 \end{bmatrix}$.
Rohini wants to give a rectangular plot of land for a school in her village. When she was asked to mention the dimensions of the plot, she told that if its length is decreased by 50 m and breadth is increased by 50 m, then its area does not alter, but if its length is decreased by 10 m and breadth i...
If $A = \begin{bmatrix} 0 & 1 & -1 \\ 1 & 2 & 1 \\ 0 & 3 & -2 \end{bmatrix}$, then the value of $|A \text{ adj } (A)|$ is : (A) $-1$ (B) $1$ (C) $2$ (D) $3$
(a) Find a matrix A such that $A \begin{bmatrix} 4 & 0 \\ -1 & -2 \end{bmatrix} = \begin{bmatrix} 17 & 10 \\ 0 & -16 \end{bmatrix}$. Also, find $A^{-1}$. OR (b) Given a square matrix A of order 3 such that $A^2 = \begin{bmatrix} 3 & -4 & 4 \\ 0 & -1 & 0 \\ -2 & 2 & -3 \end{bmatrix}$, show that $A^3 ...
If $A = \begin{bmatrix} 1 & 3 & 3 \\ 1 & 3 & 4 \\ 1 & 4 & 3 \end{bmatrix}$, then the value of $\text{det} (A^{-1})$ is :
(A) $-1$
(B) $1$
(C) $0$
(D) $2$
Find a matrix A such that $A \begin{bmatrix} 4 & 0 \\ -1 & -2 \end{bmatrix} = \begin{bmatrix} 17 & 10 \\ 0 & -16 \end{bmatrix}$. Also, find $A^{-1}$.
OR
Given a square matrix A of order 3 such that $A^2 = \begin{bmatrix} 3 & -4 & 4 \\ 0 & -1 & 0 \\ -2 & 2 & -3 \end{bmatrix}$, show that $A^3 = A^{-1}$.
If $|A| = 2$, where A is a $2 \times 2$ matrix, then $|4A^{-1}|$ equals : (a) $4$ (b) $2$ (c) $8$ (d) $\frac{1}{32}$
If $A = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$ and $(3I + 4A)(3I - 4A) = x^2I$, then the value(s) $x$ is/are :
(a) If $A = \begin{bmatrix} 1 & 2 & -2 \\ -1 & 3 & 0 \\ 0 & -2 & 1 \end{bmatrix}$ and $B^{-1} = \begin{bmatrix} 3 & -1 & 1 \\ -15 & 6 & -5 \\ 5 & -2 & 2 \end{bmatrix}$, find $(AB)^{-1}$. OR (b) Solve the following system of equations by matrix method : $x + 2y + 3z = 6$, $2x - y + z = 2$, $3x + 2y -...
If $\left| \frac{A^{-1}}{2} \right| = \frac{1}{k|A|}$, where $A$ is a $3 \times 3$ matrix, then the value of $k$ is :
(a) $\frac{1}{8}$
(b) $8$
(c) $2$
(d) $\frac{1}{2}$
Let A be a $3 \times 3$ matrix such that $|\text{adj A}| = 64$. Then $|\text{A}|$ is equal to :
(a) 8 only
(b) $-8$ only
(c) 64
(d) 8 or $-8$
If A = $\begin{bmatrix} 3 & 4 \\ 5 & 2 \end{bmatrix}$ and $2\text{A} + \text{B}$ is a null matrix, then B is equal to :
(a) $\begin{bmatrix} 6 & 8 \\ 10 & 4 \end{bmatrix}$
(b) $\begin{bmatrix} -6 & -8 \\ -10 & -4 \end{bmatrix}$
(c) $\begin{bmatrix} 5 & 8 \\ 10 & 3 \end{bmatrix}$
(d) $\begin{bmatrix}...
If $\begin{bmatrix} x & 2 \\ 3 & x - 1 \end{bmatrix}$ is a singular matrix, then the product of all possible values of x is :
(a) $6$
(b) $- 6$
(c) $0$
(d) $- 7$
The value of $|A|$, if $A = \begin{bmatrix} 0 & 2x-1 & \sqrt{x} \\ 1-2x & 0 & 2\sqrt{x} \\ -\sqrt{x} & -2\sqrt{x} & 0 \end{bmatrix}$, where $x \in \mathbb{R}^+$, is (a) $(2x+1)^2$ (b) 0 (c) $(2x+1)^3$ (d) $(2x-1)^2$
Given that $A$ is a square matrix of order 3 and $|A| = -2$, then $|adj(2A)|$ is equal to
(a) $-2^6$
(b) $+4$
(c) $-2^8$
(d) $2^8$
Using the matrix method, solve the following system of linear equations :
If $A = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}$, then $A^{2023}$ is equal to
(A) $\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}$
(B) $\begin{bmatrix} 0 & 2023 \\ 0 & 0 \end{bmatrix}$
(C) $\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$
(D) $\begin{bmatrix} 2023 & 0 \\ 0 & 2023 \end{bmatrix}$
If $(a, b)$, $(c, d)$ and $(e, f)$ are the vertices of $\Delta ABC$ and $\Delta$ denotes the area of $\Delta ABC$, then $\left| \begin{matrix} a & c & e \\ b & d & f \\ 1 & 1 & 1 \end{matrix} \right|^2$ is equal to
(A) $2\Delta^2$
(B) $4\Delta^2$
(C) $2\Delta$
(D) $4\Delta$
(b) If f($\alpha$) = $\begin{bmatrix} \cos \alpha & -\sin \alpha & 0 \\ \sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1 \end{bmatrix}$, prove that f($\alpha$) $\cdot$ f(-$\beta$) = f($\alpha$ - $\beta$)
If $A$ is a $2 \times 3$ matrix such that $AB$ and $AB'$ both are defined, then order of the matrix $B$ is
(A) $2 \times 2$
(B) $2 \times 1$
(C) $3 \times 2$
(D) $3 \times 3$
(a) If $A = \begin{bmatrix} -3 & -2 & -4 \\ 2 & 1 & 2 \\ 2 & 1 & 3 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 2 & 0 \\ -2 & -1 & -2 \\ 0 & -1 & 1 \end{bmatrix}$, then find $AB$ and use it to solve the following system of equations :
$x - 2y = 3$
$2x - y - z = 2$
$-2y + z = 3$
(b) If $f(\alpha) = \begin{bmatrix} \cos \alpha & -\sin \alpha & 0 \\ \sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1 \end{bmatrix}$, then prove that $f(\alpha) \cdot f(-\beta) = f(\alpha - \beta)$
OR
If $f(\alpha) = \begin{bmatrix} \cos \alpha & -\sin \alpha & 0 \\ \sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1 \end{bmatrix}$, prove that $f(\alpha) \cdot f(-\beta) = f(\alpha - \beta)$
In a $3 \times 3$ matrix $A = [a_{ij}]$ whose elements are given by $a_{ij} = \frac{1}{2} | -3i + j |$, the element $a_{31}$ is :
(a) $-4$
(b) $5$
(c) $4$
(d) $8$
If $2 \begin{bmatrix} 1 & 3 \\ 0 & x \end{bmatrix} + \begin{bmatrix} y & 0 \\ 1 & 2 \end{bmatrix} = \begin{bmatrix} 5 & 6 \\ 1 & 8 \end{bmatrix}$, then $2x - y$ is equal to :
(a) $3$
(b) $13$
(c) $-3$
(d) $0$
If $A = \begin{bmatrix} 1 & 2 \\ 3 & 5 \end{bmatrix}$, then $A^{-1}$ is given by :
(a) $\begin{bmatrix} 5 & -2 \\ -3 & 1 \end{bmatrix}$
(b) $\begin{bmatrix} -5 & 2 \\ 3 & -1 \end{bmatrix}$
(c) $\begin{bmatrix} -5 & 3 \\ 2 & -1 \end{bmatrix}$
(d) $\begin{bmatrix} 5 & 3 \\ 1 & 2 \end{bmatrix}$
In the determinant $\begin{vmatrix} 2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7 \end{vmatrix}$, $M_{23}$ is :
(where $M_{ij}$ denotes the minor of element $a_{ij}$)
(a) $7$
(b) $-13$
(c) $13$
(d) $-7$
Assertion (A) : Matrix A = $\begin{bmatrix} 0 & -3 & 5 \\ 3 & 0 & -2 \\ -5 & 2 & 0 \end{bmatrix}$ is a skew-symmetric matrix. Reason (R) : If $A' = -A$, then A is a skew-symmetric matrix.
Using matrices, solve the following system of linear equations : x - y + 2z = 7 ; 3x + 4y - 5z = - 5 ; 2x - y + 3z = 12
(a) If A = egin{bmatrix} 1 & 0 & 2 \ 0 & 2 & 1 \ 2 & 0 & 3 \end{bmatrix}, then show that A^{3} - 6A^{2} + 7A + 2 I = O.
OR
(b) If A = egin{bmatrix} 3 & 2 \ 5 & -7 \end{bmatrix}, then find A^{-1} and use it to solve the following system of equations :
3x + 5y = 11, 2x - 7y = -3.
If A = $\begin{bmatrix} 1 & 4 & x \\ z & 2 & y \\ -3 & -1 & 3 \end{bmatrix}$ is a symmetric matrix, then the value of x + y + z is :
(a) 10
(b) 6
(c) 8
(d) 0
If A . (adj A) = $\begin{bmatrix} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{bmatrix}$, then the value of |A| + |adj A| is equal to :
(a) 12
(b) 9
(c) 3
(d) 27
Let A be a skew-symmetric matrix of order 3. If $|A| = x$, then $(2023)^x$ is equal to :
(a) $2023$
(b) $\frac{1}{2023}$
(c) $(2023)^2$
(d) $1$
If A = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} and B = \begin{bmatrix} 1 & 1 \\ 0 & 0 \end{bmatrix}, then B'A' is equal to :
(a) \begin{bmatrix} 1 & 1 \\ 0 & 0 \end{bmatrix}
(b) \begin{bmatrix} 1 & 0 \\ 1 & 0 \end{bmatrix}
(c) \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}
(d) \begin{bmatrix} 0 &...
If $A \cdot (\text{adj } A) = \begin{bmatrix} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{bmatrix}$, then the value of $|A| + |\text{adj } A|$ is equal to :
(a) 12
(b) 9
(c) 3
(d) 27
For what value of $x \in \left[ 0, \frac{\pi}{2} \right]$, is $A + A' = \sqrt{3} \text{ I}$, where $A = \begin{bmatrix} \cos x & \sin x \\ -\sin x & \cos x \end{bmatrix}$?
(a) $\frac{\pi}{3}$
(b) $\frac{\pi}{6}$
(c) 0
(d) $\frac{\pi}{2}$
(a) If $A = \begin{bmatrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{bmatrix}$, then show that $A^{3} - 6A^{2} + 7A + 2I = O$. OR (b) If $A = \begin{bmatrix} 3 & 2 \\ 5 & -7 \end{bmatrix}$, then find $A^{-1}$ and use it to solve the following system of equations : $3x + 5y = 11$, $2x - 7y = -3$.
If $A = \begin{bmatrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{bmatrix}$, then show that $A^3 - 6A^2 + 7A + 2 I = O$.
If $A = \begin{bmatrix} 3 & 2 \\ 5 & -7 \end{bmatrix}$, then find $A^{-1}$ and use it to solve the following system of equations : $3x + 5y = 11, 2x - 7y = -3$.
If $x \begin{bmatrix} 1 \\ 2 \end{bmatrix} + y \begin{bmatrix} 2 \\ 5 \end{bmatrix} = \begin{bmatrix} 4 \\ 9 \end{bmatrix}$, then :
(a) $x = 1, \; y = 2$
(b) $x = 2, \; y = 1$
(c) $x = 1, \; y = -1$
(d) $x = 3, \; y = 2$
The value of $\begin{vmatrix} x+y & y+z & z+x \\ z & x & y \\ 1 & 1 & 1 \end{vmatrix}$ is :
(a) $0$
(b) $1$
(c) $x + y + z$
(d) $2(x + y + z)$
If $\begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 6 \\ 3 \\ 2 \end{bmatrix}$, then the value of $(2x + y - z)$ is :
(a) 1
(b) 2
(c) 3
(d) 5
The value of the determinant $\begin{vmatrix} 2 & 7 & 1 \\ 1 & 1 & 1 \\ 10 & 8 & 1 \end{vmatrix}$ is :
(a) $47$
(b) $-79$
(c) $49$
(d) $-51$
If $A$ is a $3 \times 4$ matrix and $B$ is a matrix such that $A'B$ and $AB'$ are both defined, then the order of the matrix $B$ is :
(a) $3 \times 4$
(b) $3 \times 3$
(c) $4 \times 4$
(d) $4 \times 3$
Find the inverse of the matrix $A = \begin{bmatrix} 1 & -1 & 2 \\ 0 & 2 & -3 \\ 3 & -2 & 4 \end{bmatrix}$. Using the inverse, $A^{-1}$, solve the system of linear equations
$$x - y + 2z = 1; \quad 2y - 3z = 1; \quad 3x - 2y + 4z = 3.$$
If for a square matrix $A$, $A^2 - A + I = O$, then $A^{-1}$ equals
(a) $A$
(b) $A + I$
(c) $I - A$
(d) $A - I$
If $\begin{vmatrix} \alpha & 3 & 4 \\ 1 & 2 & 1 \\ 1 & 4 & 1 \end{vmatrix} = 0$, then the value of $\alpha$ is
(a) 1
(b) 2
(c) 3
(d) 4
If $A = \begin{bmatrix} 1 & 2 & 3 \\ 3 & -2 & 1 \\ 4 & 2 & 1 \end{bmatrix}$, then show that $A^{3} - 23A - 40I = O$.
Case Study-II
Gautam buys 5 pens, 3 bags and 1 instrument box and pays a sum of ₹ 160. From the same shop, Vikram buys 2 pens, 1 bag and 3 instrument boxes and pays a sum of ₹ 190. Also Ankur buys 1 pen, 2 bags and 4 instrument boxes and pays a sum of ₹ 250.
Based on the above information, answer th...
If $A = [a_{ij}]$ is a square matrix of order 2 such that $a_{ij} = \begin{cases} 1, & \text{when } i \neq j \\ 0, & \text{when } i = j \end{cases}$, then $A^2$ is
(a) $\begin{bmatrix} 1 & 0 \\ 1 & 0 \end{bmatrix}$
(b) $\begin{bmatrix} 1 & 1 \\ 0 & 0 \end{bmatrix}$
(c) $\begin{bmatrix} 1 & 1 \\ 1 & ...
The value of the determinant $\begin{vmatrix} 6 & 0 & -1 \\ 2 & 1 & 4 \\ 1 & 1 & 3 \end{vmatrix}$ is
(a) $10$
(b) $8$
(c) $7$
(d) $-7$
Case Study-I
Gautam buys 5 pens, 3 bags and 1 instrument box and pays a sum of ₹ 160. From the same shop, Vikram buys 2 pens, 1 bag and 3 instrument boxes and pays a sum of ₹ 190. Also Ankur buys 1 pen, 2 bags and 4 instrument boxes and pays a sum of ₹ 250.
Based on the above information, answer the...
If A and B are two matrices such that $AB = A$ and $BA = B$, then $B^2$ is equal to :
(a) B
(b) A
(c) I
(d) O
If $A = \begin{bmatrix} 5 & x \\ y & 0 \end{bmatrix}$ is a symmetric matrix, then :
(a) $x = 0, y = 5$
(b) $x = 5, y = 0$
(c) $x = y$
(d) $x + y = 0$
Write the matrix $\begin{bmatrix} 7 & -3 & -3 \\ -1 & 1 & 0 \\ -1 & 0 & 1 \end{bmatrix}$ as a sum of a symmetric and a skew symmetric matrix.
10 students were selected from a school on the basis of values for giving awards and were divided into three groups. The first group comprises hard workers, the second group has honest and law abiding students, and the third group contains vigilant and obedient students. Double the number of student...
If matrix $A = \begin{bmatrix} 3 & 2 & 1 \\ 4 & 1 & 3 \\ 1 & 1 & 1 \end{bmatrix}$, find $A^{-1}$ and hence solve the following system of linear equations:
$3x + 2y + z = 2000$
$4x + y + 3z = 2500$
$x + y + z = 900$
If $\begin{bmatrix} 3 & 2 \\ 1 & x \end{bmatrix} \begin{bmatrix} x \\ 1 \end{bmatrix} = \begin{bmatrix} 14 \\ 8 \end{bmatrix}$, then x is :
(a) $\frac{16}{3}$
(b) $-3$
(c) $-4$
(d) $4$
Let $A = \begin{bmatrix} 1 & 1 & 1 \\ 6 & 7 & 8 \\ 6 & 7 & -8 \end{bmatrix}$, find $A^{-1}$ and hence solve the following system of linear equations :
\begin{align*}
x + y + z &= 5000 \\
6x + 7y + 8z &= 35800 \\
6x + 7y - 8z &= 7000
\end{align*}
In the matrix equation $\begin{bmatrix} x + y + z \\ x + z \\ y + 2z \end{bmatrix} = \begin{bmatrix} 3 \\ 2 \\ -1 \end{bmatrix}$, the value of z is :
(a) 1
(b) 2
(c) -1
(d) -2
If $A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 0 & 2 \\ 3 & 1 & 1 \end{bmatrix}$, find $A^{-1}$ and hence solve the following system of linear equations:
$x + y + z = 6$
$x + 2z = 7$
$3x + y + z = 12$
If the matrix $\begin{bmatrix} 0 & 3 & 5 \\ k+1 & 0 & 4 \\ -5 & k & 0 \end{bmatrix}$ is a skew symmetric matrix, then the value of k is : (a) 4 (b) 2 (c) -2 (d) -4
If A = $\begin{bmatrix} 4 & -3 \\ 9 & -3 \end{bmatrix}$, then $|A|$ is equal to : (a) 15 (b) -42 (c) 0 (d) 25
If A = $\begin{bmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{bmatrix}$ and $A+A' = I$, then the value of $\alpha$ is : (a) $\frac{\pi}{6}$ (b) $\frac{\pi}{3}$ (c) $\frac{\pi}{2}$ (d) $\frac{\pi}{4}$
If A is a $2 \times 3$ matrix and B is another matrix such that both $A'B$ and $BA'$ are defined, then order of B is : (a) $3 \times 2$ (b) $2 \times 3$ (c) $3 \times 3$ (d) $2 \times 2$
Express the matrix $A = egin{bmatrix} 3 & -4 \ 1 & -1 \end{bmatrix}$ as the sum of a symmetric matrix and a skew-symmetric matrix.
If A, B are non-singular square matrices of the same order, then $(AB^{-1})^{-1} =$
(a) $A^{-1}B$ (b) $A^{-1}B^{-1}$ (c) $BA^{-1}$ (d) $AB$
If $\begin{vmatrix} 2 & 4 \\ 5 & 1 \end{vmatrix} = \begin{vmatrix} 2x & 4 \\ 6 & x \end{vmatrix}$, then the possible value(s) of 'x' is/are
(a) $3$ (b) $\sqrt{3}$ (c) $-\sqrt{3}$ (d) $\sqrt{3}, -\sqrt{3}$
If $A = \begin{bmatrix} 2 & -3 & 5 \\ 3 & 2 & -4 \\ 1 & 1 & -2 \end{bmatrix}$, find $A^{-1}$. Use $A^{-1}$ to solve the following system of equations
$2x - 3y + 5z = 11, 3x + 2y - 4z = -5, x + y - 2z = -3$