If A = bmatrix 0 & 0 & √3 \\ 0 & √3 & 0 \\ √3 & 0 & 0 bmatrix, then, adj A, is equal to Option 1: 3 Option 2: 9 Option 3: 27 Option 4: 81

Answer: 27. Solution: For any n x n matrix A:, adj A, =, A, ^(n-1) Since A is a 3 x 3 matrix:, adj A, =, A, ^(3-1) =, A, ^2 Given: A = bmatrix 0 & 0 & √3 \\ 0 & √3 & 0 \\ √3 & 0 & 0 bmatrix Expanding along the first row:, A, = 0 x C_11 + 0 x C_12 + √3 x C_13 Finding the cofactor C_13 by removing row 1 and column 3: C_13 = (-1)^(1+3) x vmatrix 0 & √3 \\ √3 & 0 vmatrix = (+1) x (0 x 0 - √3 x √3) = 1 x (0 - 3) = -3 Therefore:, A, = √3 x (-3) = -3√3, adj A, =, A, ^2 = (-3√3)^2 = (-3)^2 x (√3)^2 = 9 x 3 = 27

Algebra

Matrices & Determinants

Medium

common

If $A = \begin{bmatrix} 0 & 0 & \sqrt{3} \\ 0 & \sqrt{3} & 0 \\ \sqrt{3} & 0 & 0 \end{bmatrix}$ , then $|adj A|$ is equal to

✅ Correct Option: 3

For any $n \times n$ matrix $A$ :

$|adj \, A| = |A|^{n-1}$

Since $A$ is a $3 \times 3$ matrix:

$|adj \, A| = |A|^{3-1}$

$= |A|^2$

Given: $A = \begin{bmatrix} 0 & 0 & \sqrt{3} \\ 0 & \sqrt{3} & 0 \\ \sqrt{3} & 0 & 0 \end{bmatrix}$

Expanding along the first row:

$|A| = 0 \times C_{11} + 0 \times C_{12} + \sqrt{3} \times C_{13}$

Finding the cofactor $C_{13}$ by removing row 1 and column 3:

$C_{13} = (-1)^{1+3} \times \begin{vmatrix} 0 & \sqrt{3} \\ \sqrt{3} & 0 \end{vmatrix}$

$= (+1) \times (0 \times 0 - \sqrt{3} \times \sqrt{3})$

$= 1 \times (0 - 3)$

$= -3$

Therefore:

$|A| = \sqrt{3} \times (-3)$

$= -3\sqrt{3}$

$|adj \, A| = |A|^2$

$= (-3\sqrt{3})^2$

$= (-3)^2 \times (\sqrt{3})^2$

$= 9 \times 3$

$= 27$

2025: 13 May Shift 1

If A and B are invertible matrices of same order, then which one of the following is NOT true?

2025: 13 May Shift 2

Let $A = \begin{bmatrix} 2 & -3 & 4 \\ 0 & 1 & 5 \\ -4 & 2 & 3 \end{bmatrix}$ and $a_{ij}$ be any element of matrix A, i, j ∈ {1,2,3}, then which of the following are TRUE?

(A) Minor of $a_{23} = 16$

(B) Minor of $a_{23} = -8$

(D) Cofactor of $a_{23} = 8$

(E) Cofactor of $a_{13} = 4$

Choose the correct answer from the options given below:

2025: 3 June Shift 1

The value of k for which the system of equations

$x + y + z = 1$

$x - ky + z = 1$

$x - y + z = 1$

has more than one solutions is

2026: 22 May Shift 2

Match List-I with List-II

List-I	List-II
(A) If $A = [a_{ij}]_{n \times n}$ , where $a_{ij} = \begin{cases}0, & i \ne j \\ k, & i=j, k\ne 0\end{cases}$ then A is ___	(I) Symmetric matrix
(B) If $A = [a_{ij}]_{2\times 2}$ , where $a_{ij} = \begin{cases}1, & i \ne j \\ 0, & i=j\end{cases}$ , then $A^2$ is ___	(II) Skew-Symmetric matrix
(C) If A and B are matrices of the same order, then $(AB^T - BA^T)$ is ___	(III) Identity matrix
(D) If A and B are symmetric matrices of the same order, then $(AB+BA)$ is ___	(IV) Scalar matrix

Choose the correct answer from the options given below:

2026: 29 May Shift 2

Let $A=[a_{ij}]_{3\times3}$ , where $a_{ij}=i-j$ , then which of the following statements is/are TRUE ?

A. A is diagonal matrix

B. A is square matrix

C. A is symmetric matrix

D. A is skew symmetric matrix

Choose the correct answer from the options given below:

2026: 14 May Shift 2

If $(AB)^T = C$ , where $A = \begin{bmatrix} x & -1 \\ 2 & 3 \end{bmatrix}$ , $B = \begin{bmatrix} 4 & 5 \\ 1 & -2 \end{bmatrix}$ and $C = \begin{bmatrix} 3 & 11 \\ 7 & y \end{bmatrix}$ , then $x+y$ is :

If A=[003030300]A = \begin{bmatrix} 0 & 0 & \sqrt{3} \\ 0 & \sqrt{3} & 0 \\ \sqrt{3} & 0 & 0 \end{bmatrix}A=​003​​03​0​3​00​​, then ∣adjA∣|adj A|∣adjA∣ is equal to

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

If $A = \begin{bmatrix} 0 & 0 & \sqrt{3} \\ 0 & \sqrt{3} & 0 \\ \sqrt{3} & 0 & 0 \end{bmatrix}$ , then $|adj A|$ is equal to