Algebra

Matrices & Determinants

Medium

common

If A and B are symmetric matrices, then AB - BA is

Singular matrix

Zero matrix

Symmetric matrix

Skew symmetric matrix

✅ Correct Option: 4

Given: A and B are symmetric matrices

A symmetric matrix equals its own transpose: $A = A^T$ and $B = B^T$

Let $M = AB - BA$

Taking the transpose:

$M^T = (AB - BA)^T$

$M^T = (AB)^T - (BA)^T$

Using the transpose property $(PQ)^T = Q^T P^T$ :

$M^T = B^T A^T - A^T B^T$

Since A and B are symmetric, $A^T = A$ and $B^T = B$ :

$M^T = BA - AB$

$M^T = -(AB - BA)$

$M^T = -M$

Since $M^T = -M$ , the matrix $AB - BA$ is a skew-symmetric matrix.

2026: 30 May Shift 2

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(A) $|A| = 7$

(B) minor of $3 = -5$

(D) $adj(A) = \begin{bmatrix} 3 & -2 \\ -4 & 5 \end{bmatrix}$

Choose the correct answer from the options given below:

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2023: 22 May Shift 3

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If A and B are symmetric matrices, then AB - BA is

Solution

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