Algebra

Matrices & Determinants

Medium

applied

If A and B are symmetric matrices of same order, then which of the following are correct?

(A) AB-BA is a skew-symmetric matrix.

(B) AB+BA is a skew-symmetric matrix.

(C) $AB^T-BA^T$ is a skew-symmetric matrix.

(D) AB+BA is a symmetric matrix.

Choose the correct answer from the options given below:

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

(A), (B) and (C) only

(A) and (D) only

✅ Correct Option: 1

Given that A and B are symmetric matrices of the same order.

For symmetric matrices: $A^T = A$ and $B^T = B$

Checking statement (A): AB - BA

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

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

$= BA - AB$

$= -(AB - BA)$

Therefore, AB - BA is skew-symmetric.

Checking statement (B): AB + BA

$(AB + BA)^T = (AB)^T + (BA)^T$

$= B^T A^T + A^T B^T$

$= BA + AB$

$= AB + BA$

Therefore, AB + BA is symmetric, not skew-symmetric.

Checking statement (C): $AB^T - BA^T$

Since B is symmetric: $B^T = B$

Since A is symmetric: $A^T = A$

$AB^T - BA^T = AB - BA$

From statement (A), AB - BA is skew-symmetric.

Therefore, $AB^T - BA^T$ is skew-symmetric.

Checking statement (D): AB + BA

From statement (B), $(AB + BA)^T = AB + BA$

Therefore, AB + BA is symmetric.

Statements (A), (C), and (D) are correct.

2026: 18 May Shift 1

Let $A$ and $B$ be two symmetric matrices of same order, then

Match List-I with List-II

List-I	List-II
(A). $AB - BA$	(I). Neither symmetric nor Skew symmetric
(B). $BA - 2AB$	(II). Skew symmetric
(C). $AB + BA$	(III). Null matrix
(D). $(AB)^T - BA$	(IV). Symmetric

Choose the correct answer from the options given below:

2026: 18 May Shift 2

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Solution

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