For a matrix A = bmatrix 1 & 4 & 5 \\ 3 & 2 & 6 \\ 0 & 1 & 0 bmatrix, where I is identity matrix of order 3 Match List-I with List-II, List-I, List-II, (A) A, (I) 72, (B) (adjA)A, (II) 18, (C) 2A, (III) 9, (D) 2 A, (IV) 9I, Choose the correct answer from the options given below: Option 1: (A) - (III), (B) - (IV), (C) - (I), (D) - (II) Option 2: (A) - (III), (B) - (II), (C) - (I), (D) - (IV) Option 3: (A) - (II), (B) - (IV), (C) - (III), (D) - (I) Option 4: (A) - (II), (B) - (IV), (C) - (I), (D) - (III)

Answer: (A) - (III), (B) - (IV), (C) - (I), (D) - (II). Solution: Expanding along the third row, A, =-1 (1* 6-5* 3)+0=9, so (A)-(III). For any square matrix (adj A)A=, A, I=9I, so (B)-(IV). For order 3, 2A, =2^3, A, =8x 9=72, so (C)-(I). Finally 2, A, =18, so (D)-(II).

Algebra

Matrices & Determinants

Medium

common

For a matrix $A = \begin{bmatrix} 1 & 4 & 5 \\ 3 & 2 & 6 \\ 0 & 1 & 0 \end{bmatrix}$ , where $I$ is identity matrix of order 3

Match List-I with List-II

List-I List-II
(A) $\left\vert A \right\vert$ (I) $72$
(B) $(\text{adj}A)A$ (II) $18$
(C) $\left\vert 2A \right\vert$ (III) $9$
(D) $2\left\vert A \right\vert$ (IV) $9I$

Choose the correct answer from the options given below:

List-I	List-II
(A) $\left\vert A \right\vert$	(I) $72$
(B) $(\text{adj}A)A$	(II) $18$
(C) $\left\vert 2A \right\vert$	(III) $9$
(D) $2\left\vert A \right\vert$	(IV) $9I$

(A) - (III), (B) - (IV), (C) - (I), (D) - (II)

(A) - (III), (B) - (II), (C) - (I), (D) - (IV)

(A) - (II), (B) - (IV), (C) - (III), (D) - (I)

(A) - (II), (B) - (IV), (C) - (I), (D) - (III)

✅ Correct Option: 1

Expanding along the third row, $|A|=-1\,(1\cdot 6-5\cdot 3)+0=9$ , so (A)-(III). For any square matrix $(adj\,A)A=|A|I=9I$ , so (B)-(IV). For order 3, $|2A|=2^3|A|=8\times 9=72$ , so (C)-(I). Finally $2|A|=18$ , so (D)-(II).

2025: 14 May Shift 2

If $A = \begin{bmatrix} 2 & 1 & -1 \\ 0 & 1 & 2 \\ 2 & -1 & \lambda \end{bmatrix}$ is a singular matrix, then the value of $\lambda$ is

2026: 15 May Shift 1

Match the LIST-I with LIST-II

	LIST-I		LIST-II
A.	$A = \begin{bmatrix}3 & 5\\ 5 & 2\end{bmatrix}$	I.	Scalar matrix
B.	$A = \begin{bmatrix}0 & -2\\ 2 & 0\end{bmatrix}$	II.	Upper triangular matrix
C.	$A = \begin{bmatrix}5 & 3\\ 0 & 4\end{bmatrix}$	III.	Symmetric matrix
D.	$A = \begin{bmatrix}3 & 0\\ 0 & 3\end{bmatrix}$	IV.	Skew symmetric matrix

Choose the correct answer from the options given below:

2026: 29 May Shift 2

Match the LIST-I with LIST-II

LIST-I	LIST-II
A. $\begin{vmatrix} x+2 & 3 \\ x+5 & 4 \end{vmatrix} = 3$	I. $x = 10$
B. $\begin{vmatrix} x+2 & x+5 \\ 3 & 4 \end{vmatrix} = 13$	II. $x = 60$
C. $\begin{vmatrix} x+4 & x-4 \\ 8 & 7 \end{vmatrix} = 0$	III. $x = 0$
D. $\begin{vmatrix} x+4 & 8 \\ x-4 & 7 \end{vmatrix} = 60$	IV. $x = 20$

Choose the correct answer from the options given below:

2025: 30 May Shift 1

If $A = \begin{bmatrix} 0 & 1 & 3 \\ 1 & 2 & x \\ 2 & 3 & 1 \end{bmatrix}$ and $A^{-1} = \begin{bmatrix} \frac{1}{2} & -4 & \frac{5}{2} \\ -\frac{1}{2} & 3 & -\frac{3}{2} \\ \frac{1}{2} & y & \frac{1}{2} \end{bmatrix}$ , then the value of $8x + 5y$ is:

2022: 16 July Shift 2

The values of $x$ and $y$ in the equation $2\begin{bmatrix} x & 1 \\ 4 & -3 \end{bmatrix} + 3\begin{bmatrix} -2 & 1 \\ 2 & y-2 \end{bmatrix} = \begin{bmatrix} 2 & 5 \\ 14 & 3 \end{bmatrix}$ are respectively:

2025: 29 May Shift 2

If the system of equation

$x - y + z = 4$

$x - 2y - 2z = 9$

$2x + y + \lambda z = 1$

has a unique solution, then

Solution

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