Algebra

Matrices & Determinants

Medium

applied

If $A = \begin{bmatrix}0 & x^2-6 & -3\\-x & 0 & -8\\x^2-2x & 8 & 0\end{bmatrix}$ is a skew symmetric matrix, then the value(s) of x is/ are -

(A) 3

(B) -3

(C) -2

(D) -1

Choose the correct answer from the options given below:

(A) and (C) only

(A), (C) and (D) only

(A) only

(B), (C) and (D) only

✅ Correct Option: 3

A skew-symmetric matrix satisfies $A^T = -A$ , which means $a_{ij} = -a_{ji}$ for all elements. The diagonal elements are always zero.

Given: $A = \begin{bmatrix}0 & x^2-6 & -3\\-x & 0 & -8\\x^2-2x & 8 & 0\end{bmatrix}$

For A to be skew-symmetric, the following conditions must hold:

$a_{12} = -a_{21}$

$a_{13} = -a_{31}$

$a_{23} = -a_{32}$

From $a_{12} = -a_{21}$ :

$x^2 - 6 = -(-x)$

$x^2 - 6 = x$

$x^2 - x - 6 = 0$

$(x - 3)(x + 2) = 0$

$x = 3$ or $x = -2$

From $a_{13} = -a_{31}$ :

$-3 = -(x^2 - 2x)$

$-3 = -x^2 + 2x$

$x^2 - 2x - 3 = 0$

$(x - 3)(x + 1) = 0$

$x = 3$ or $x = -1$

From $a_{23} = -a_{32}$ :

$-8 = -(8)$

$-8 = -8$

This condition is always satisfied.

For the matrix to be skew-symmetric, all three conditions must be satisfied simultaneously.

From the first condition: $x = 3$ or $x = -2$

From the second condition: $x = 3$ or $x = -1$

The only value common to both conditions is $x = 3$ .

Therefore, the correct answer is (A) only.

2025: 15 May Shift 2

If $A = \begin{bmatrix} 2 & 0 & 3 \\ -1 & 1 & 3 \\ 0 & -4 & 0 \end{bmatrix}$ , then the value of det (2A) is

2025: 15 May Shift 1

The value of $\begin{vmatrix} 1 & x & y \\ 1 & x+y & y \\ 1 & x & x+y \end{vmatrix}$ is

2025: 13 May Shift 1

If the points $(2, -3)$ , $(\lambda, -1)$ and $(0, 4)$ are collinear, then the value of $\lambda$ is

2026: 29 May Shift 1

Which of the following are correct?

(A) If $A=\begin{bmatrix}7 & 5 \\ 4 & 3\end{bmatrix}$ , then $A^{-1}=\begin{bmatrix}3 & -5 \\ -4 & 7\end{bmatrix}$

(B) If $A=\begin{bmatrix}3 & 5 \\ 6 & 10\end{bmatrix}$ , then $A$ is a non-singular matrix

(D) If $A=\begin{bmatrix}6 & x \\ y & 0\end{bmatrix}$ and $A=A^T$ , then $x=y$

Choose the correct answer from the options given below:

2026: 19 May Shift 2

Match the LIST-I with LIST-II

	LIST-I Characteristic of square matrix		LIST-II Type of matrix
A.	A is a square matrix such that $A + A^T = 0$ .	I.	Identity matrix or scalar matrix
B.	B is a square matrix such that $B^2 = B$	II.	Diagonal matrix
C.	C is a square matrix with all equal diagonal element and all the non-diagonal elements are zero.	III.	Skew symmetric matrix
D.	D is a matrix such that all the non-diagonal elements are zero.	IV.	Idempotent matrix

Choose the correct answer from the options given below:

2026: 21 May Shift 2

If matrix $A = \begin{bmatrix}12 & 7 & 9\\6 & 3 & 2\\5 & 1 & 8\end{bmatrix}$ , then

Match the LIST-I with LIST-II

	LIST-I		LIST-II
A.	$\vert A \vert$	I.	$38$
B.	Cofactor of $a_{31}$	II.	$-23$
C.	Minor of $a_{23}$	III.	$-83$
D.	Cofactor of $a_{12}$	IV.	$-13$

Choose the correct answer from the options given below:

Solution

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