Question

Let $A$ be square matrix. Then which of the following is not a symmetric matrix.

• A. $A+A^{\prime }$
• B. $A{A}^{\prime }$
• C. $A^{\prime }A$
• D. $A-A^{\prime }$

Answer 1

Answer: (D)

$A-A^{\prime }$

If $A$ is a square matrix, then and if $A^{\prime }$ represents its transpose, then
$A+A^{\prime }$ is symmetric and $A-A^{\prime }$ is skew symmetric.
Hence matrix A can be written as
$A=\left(\frac{A+A^{\prime }}{2}\right)+\left(\frac{A-A^{\prime }}{2}\right)$
Therefore of all the above matrix,
$A-A^{\prime }$ is not symmetric.