If the matrix M = bmatrix 0 & -1 & 3 \\ 1 & & -5 \\ -6 & 5 & 0 bmatrix is skew-symmetric, then Option 1: = 2, = 1 Option 2: = 2, = -1 Option 3: = 2, = 0 Option 4: = 1, = 0

Answer: = 2, = 0. Solution: A matrix M is skew-symmetric when M^T = -M, which means all diagonal elements must be 0 and elements are opposite across the diagonal. Given matrix: M = bmatrix 0 & -1 & 3 \\ 1 & & -5 \\ -6 & 5 & 0 bmatrix The diagonal elements are: 0, 0 For skew-symmetric, all diagonal elements must be 0. Therefore = 0 For skew-symmetric matrices, the element at position (i,j) equals the negative of the element at position (j,i). Comparing position (1,3) and (3,1): Position (1,3) = 3 Position (3,1) = -6 For skew-symmetric: 3 = -(-6) 3 = 6 = 2 Comparing position (1,2) and (2,1): Position (1,2) = -1 Position (2,1) = 1 This satisfies -1 = -(1), confirming our result. Therefore, = 2 and = 0.

Algebra

Matrices & Determinants

Medium

applied

If the matrix $M = \begin{bmatrix} 0 & -1 & 3\alpha \\ 1 & \beta & -5 \\ -6 & 5 & 0 \end{bmatrix}$ is skew-symmetric, then

$\alpha = 2, \beta = 1$

$\alpha = 2, \beta = -1$

$\alpha = 2, \beta = 0$

$\alpha = 1, \beta = 0$

✅ Correct Option: 3

A matrix $M$ is skew-symmetric when $M^T = -M$ , which means all diagonal elements must be $0$ and elements are opposite across the diagonal.

Given matrix: $M = \begin{bmatrix} 0 & -1 & 3\alpha \\ 1 & \beta & -5 \\ -6 & 5 & 0 \end{bmatrix}$

The diagonal elements are: $0, \beta, 0$

For skew-symmetric, all diagonal elements must be $0$ .

Therefore $\beta = 0$

For skew-symmetric matrices, the element at position $(i,j)$ equals the negative of the element at position $(j,i)$ .

Comparing position $(1,3)$ and $(3,1)$ :

Position $(1,3) = 3\alpha$

Position $(3,1) = -6$

For skew-symmetric: $3\alpha = -(-6)$

$3\alpha = 6$

$\alpha = 2$

Comparing position $(1,2)$ and $(2,1)$ :

Position $(1,2) = -1$

Position $(2,1) = 1$

This satisfies $-1 = -(1)$ , confirming our result.

Therefore, $\alpha = 2$ and $\beta = 0$ .

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(A) $a=1, b=1$

(B) $a=0, b=1$

(D) $a\neq0$ and $b\in(3,4)$

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Match List-I with List-II

List-I	List-II
(Matrix)	(Determinant)
(A) $\begin{bmatrix} 1 & 7 \\ -3 & 5 \end{bmatrix}$	(I) 24
(B) $\begin{bmatrix} -2 & 5 \\ -3 & -3 \end{bmatrix}$	(II) 32
(C) $\begin{bmatrix} -12 & 8 \\ -16 & 8 \end{bmatrix}$	(III) 21
(D) $\begin{bmatrix} 15 & 9 \\ -21 & -11 \end{bmatrix}$	(IV) 26

Choose the correct answer from the options given below:

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Let the matrix $A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}$ . Then which of the following are true?

(A) adj $A = \begin{bmatrix} 4 & -3 \\ -1 & 2 \end{bmatrix}$

(B) det $(A) = 5$

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If the matrix M=[0−13α1β−5−650]M = \begin{bmatrix} 0 & -1 & 3\alpha \\ 1 & \beta & -5 \\ -6 & 5 & 0 \end{bmatrix}M=​01−6​−1β5​3α−50​​ is skew-symmetric, then

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CUET Mathematics Past Year Questions (Topic-Wise)

If the matrix $M = \begin{bmatrix} 0 & -1 & 3\alpha \\ 1 & \beta & -5 \\ -6 & 5 & 0 \end{bmatrix}$ is skew-symmetric, then