CUET Mathematics 2024 - If A and B are symmetric matrices of the same order, then A B-B A is a : | PYQs + Solutions

CUET Mathematics 2024

Algebra

Matrices & Determinants

Easy

If $A$ and $B$ are symmetric matrices of the same order, then $A B-B A$ is a :

symmetric matrix

zero matrix

skew symmetric matrix

identity matrix

✅ Correct Option: 3

A

and

B

are symmetric matrices of the same order, then

AB-BA

is a skew-symmetric matrix.

Recall that a symmetric matrix

X

has the property that

X = X^T

, where

X^T

is the transpose of

X

A skew-symmetric matrix

Y

has the property that

Y = -Y^T

Let's find the transpose of

AB - BA

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

For any matrices

P

and

Q

, we know that

(PQ)^T = Q^T P^T

. So:

(AB)^T = B^T A^T

(BA)^T = A^T B^T

Since

A

and

B

are symmetric matrices,

A = A^T

and

B = B^T

Substituting into our transpose expression:

(AB - BA)^T = B^T A^T - A^T B^T = BA - AB = -(AB - BA)

Since

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

, by definition

AB - BA

is skew-symmetric.

Note: This means all diagonal elements will be zero, and corresponding off-diagonal elements will be negatives of each other.