Question

If A and B are symmetric matrices of the same order, then AB^T – BA^T is a
(a) skew-symmetric matrix
(b) null matrix
(c) symmetric matrix
(d) none of these

Answer 1

It is given that, A and B are symmetric matrices of the same order.

∴ A^T = A and B^T = B .....(1)

Now,

${\left(A{B}^T-B{A}^T\right)}^T$

$={\left(AB-BA\right)}^T$ [Using (1)]

$={\left(AB\right)}^T-{\left(BA\right)}^T\left[{\left(X+Y\right)}^T=X^T+Y^T\right]$

$=B^T{A}^T-A^T{B}^T\left[{\left(XY\right)}^T=Y^T{X}^T\right]$

$=BA-AB$ [Using (1)]

$=-\left(A{B}^T-B{A}^T\right)$ [Using (1)]

We know that, a matrix X is skew-symmetric if $X^T=-X$.

Since ${\left(A{B}^T-B{A}^T\right)}^T=-\left(A{B}^T-B{A}^T\right)$, therefore, AB^T – BA^T is a skew-symmetric matrix.

Hence, the correct answer is option (a).