If A = bmatrix 0 & 0 & √7 \\ 0 & √7 & 0 \\ √7 & 0 & 0 bmatrix, then, adj A, is equal to Option 1: 7 Option 2: 49 Option 3: 343 Option 4: 0

Answer: 343. 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), adj A, =, A, ^2 Given: A = bmatrix 0 & 0 & √7 \\ 0 & √7 & 0 \\ √7 & 0 & 0 bmatrix Expanding along Row 1:, A, = 0 * C_11 - 0 * C_12 + √7 * C_13, A, = √7 * C_13 The cofactor C_13 is given by: C_13 = (-1)^(1+3) x M_13 where M_13 is the minor obtained by removing row 1 and column 3: M_13 = vmatrix 0 & √7 \\ √7 & 0 vmatrix M_13 = (0)(0) - (√7)(√7) M_13 = 0 - 7 M_13 = -7 Therefore: C_13 = (-1)^4 x (-7) C_13 = 1 x (-7) C_13 = -7 Substituting back:, A, = √7 x (-7), A, = -7√7 Using the formula:, adj A, =, A, ^2, adj A, = (-7√7)^2, adj A, = (-7)^2 x (√7)^2, adj A, = 49 x 7, adj A, = 343

Algebra

Matrices & Determinants

Medium

common

If $A = \begin{bmatrix} 0 & 0 & \sqrt{7} \\ 0 & \sqrt{7} & 0 \\ \sqrt{7} & 0 & 0 \end{bmatrix}$ , then $|\text{adj } A|$ is equal to

343

✅ Correct Option: 3

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

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

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

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

$|\text{adj } A| = |A|^2$

Given: $A = \begin{bmatrix} 0 & 0 & \sqrt{7} \\ 0 & \sqrt{7} & 0 \\ \sqrt{7} & 0 & 0 \end{bmatrix}$

Expanding along Row 1:

$|A| = 0 \cdot C_{11} - 0 \cdot C_{12} + \sqrt{7} \cdot C_{13}$

$|A| = \sqrt{7} \cdot C_{13}$

The cofactor $C_{13}$ is given by:

$C_{13} = (-1)^{1+3} \times M_{13}$

where $M_{13}$ is the minor obtained by removing row 1 and column 3:

$M_{13} = \begin{vmatrix} 0 & \sqrt{7} \\ \sqrt{7} & 0 \end{vmatrix}$

$M_{13} = (0)(0) - (\sqrt{7})(\sqrt{7})$

$M_{13} = 0 - 7$

$M_{13} = -7$

Therefore:

$C_{13} = (-1)^{4} \times (-7)$

$C_{13} = 1 \times (-7)$

$C_{13} = -7$

Substituting back:

$|A| = \sqrt{7} \times (-7)$

$|A| = -7\sqrt{7}$

Using the formula:

$|\text{adj } A| = |A|^2$

$|\text{adj } A| = (-7\sqrt{7})^2$

$|\text{adj } A| = (-7)^2 \times (\sqrt{7})^2$

$|\text{adj } A| = 49 \times 7$

$|\text{adj } A| = 343$

2025: 15 May Shift 1

Which of the following are correct?

(A) If A and B are symmetric matrices such that AB = BA, then AB is symmetric.

(B) If A and B are symmetric matrices of the same order, then (A+B) is a symmetric matrix.

(D) If A and B are symmetric matrices of the same order, then (AB+BA) is a skew symmetric matrix

Choose the correct answer from the options given below:

2026: 18 May Shift 2

If $x,y,z$ are non-zero real numbers such that $\begin{vmatrix} 10+x & 10 & 10 \\ 10 & 10+y & 10 \\ 10 & 10 & 10+z \end{vmatrix}=0$ , then the value of $x^{-1}+y^{-1}+z^{-1}$ is

2026: 31 May Shift 2

The number of all possible matrices of order $2 \times 2$ with each entry 0 or 1 having their determinants as positive value is :

2026: 30 May Shift 1

If the matrix $\begin{bmatrix}0 & \alpha & 2\\2 & \beta & 1\\\gamma & -1 & 0\end{bmatrix}$ is skew-symmetric, then $(\beta-\alpha-\gamma)$ equals

2025: 15 May Shift 2

Let $A = [a_{ij}]$ be a square matrix, where $a_{ij} = \begin{cases} 0, & \text{when } i = j \\ 1, & \text{otherwise} \end{cases}$ . If |adj A| = |A|², then which of the following statements are correct?

(A) A is a skew symmetric matrix.

(B) A is a non-singular matrix.

(D) A is a symmetric matrix.

Choose the correct answer from the options given below:

2025: 14 May Shift 2

If A is a square matrix of order 3 such that $|A| = 3$ , then $|adj(adj A)|$ is equal to:

If A=[007070700]A = \begin{bmatrix} 0 & 0 & \sqrt{7} \\ 0 & \sqrt{7} & 0 \\ \sqrt{7} & 0 & 0 \end{bmatrix}A=​007​​07​0​7​00​​, then ∣adj A∣|\text{adj } A|∣adj A∣ is equal to

Solution

Related questions:

Keyboard Shortcuts

CUET Mathematics Past Year Questions (Topic-Wise)

If $A = \begin{bmatrix} 0 & 0 & \sqrt{7} \\ 0 & \sqrt{7} & 0 \\ \sqrt{7} & 0 & 0 \end{bmatrix}$ , then $|\text{adj } A|$ is equal to