Algebra

Matrices & Determinants

Medium

core

If A and B are skew-symmetric matrices, then which of the following is not true?

$A^5 + B^7$ is skew-symmetric

$A^{21}$ is skew-symmetric

$B^{18}$ is symmetric

$A^6 + B^7$ is symmetric

✅ Correct Option: 4

A matrix $A$ is skew-symmetric if $A^T = -A$ .

When transposing a skew-symmetric matrix, the result is the negative of the original matrix.

For a skew-symmetric matrix $A$ raised to power $n$ :

$(A^n)^T = (-A)^n$

$(A^n)^T = (-1)^n \times A^n$

When $n$ is odd: $(A^n)^T = -A^n$ (skew-symmetric)

When $n$ is even: $(A^n)^T = A^n$ (symmetric)

For $A^5 + B^7$ :

$A^5$ has odd power, so $A^5$ is skew-symmetric.

$B^7$ has odd power, so $B^7$ is skew-symmetric.

The sum of two skew-symmetric matrices is skew-symmetric.

For $A^{21}$ :

$A^{21}$ has odd power, so $A^{21}$ is skew-symmetric.

For $B^{18}$ :

$B^{18}$ has even power, so $B^{18}$ is symmetric.

For $A^6 + B^7$ :

$A^6$ has even power, so $A^6$ is symmetric.

$B^7$ has odd power, so $B^7$ is skew-symmetric.

The sum of a symmetric matrix and a skew-symmetric matrix is neither symmetric nor skew-symmetric.

Therefore, the statement that $A^6 + B^7$ is symmetric is not true.

2022: 17 Aug Shift 2

The number of possible matrices of order $2 \times 2$ with each entry $0$ or $1$ or $2$ is :

2025: 30 May Shift 1

Let $A = [a_{ij}]_{2 \times 4}$ and $B = [b_{ij}]_{4 \times 2}$ , then $|3AB|$ is equal to

2026: 14 May Shift 2

If $A = \begin{bmatrix} 0 & x & x \\ 2y & y & -y \\ z & -z & z \end{bmatrix}$ such that $AA^T=2I$ (where I is an identity matrix of order 3), then which of the following are True?

A. $x^2=1, 3y^2=2$

B. $6y^2=4, 2z^2=3$

C. $9z^2=6, x^2=1$

D. $x^2:y^2:z^2 = 3:1:2$

Choose the correct answer from the options given below:

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: 14 May Shift 2

If $A=\begin{bmatrix}a & 1 & c\\ b & 2 & 1\\ 2 & 1 & c\end{bmatrix}$ is a singular matrix, then

2025: 26 May Shift 2

If $A = \begin{bmatrix} x+z & 2 & -3 \\ x & 0 & 4 \\ 3 & x-y & 0 \end{bmatrix}$ is a skew-symmetric matrix, then which of the following are true?

(A) $y > z > x$

(B) $x > y$

(D) $z > x$

Choose the correct answer from the options given below:

If A and B are skew-symmetric matrices, then which of the following is not true?

Solution

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