Question

If $A$ and $B$ are two symmetric matrices of the same order. Then, the matrix $AB-BA$ is equal to

• A. a symmetric matrix
• B. a skew-symmetric matrix
• C. a null matrix
• D. the identity matrix

Answer 1

The correct option is B

a skew-symmetric matrix

Explanation for the correct option:

Find the value of $AB-BA$:

Given, $A$ and $B$ are two symmetric matrices.

$\Rightarrow$$A=A\text{'}$ and $B=B\text{'}$

$\Rightarrow$$\begin{array}{rcl}(AB–BA)\text{'}& =& \left(AB\right)\text{'}–\left(BA\right)\text{'}\end{array}$

$\begin{array}{rcl}& =& B\text{'}A\text{'}–A\text{'}B\text{'}\\ & =& BA–AB\\ & =& –(AB–BA)\end{array}$

$\therefore$$AB–BA$ is a skew-symmetric matrix.

Hence, Option ‘B’ is Correct.