Question

State True or False:
$A{A}^{\prime }$ is always a symmetric matrix for any matrix $A$.

• A. True
• B. False

Answer 1

The correct option is A True

Taking transpose of $A{A}^{\prime }$
${\left(A{A}^{\prime }\right)}^{\prime }={\left(A^{\prime }\right)}^{\prime }{A}^{\prime }$ $[{\left(AB\right)}^{\prime }=B^{\prime }{A}^{\prime }]$
$\Rightarrow {\left(AA\right)}^{\prime }=A{A}^{\prime }$
(If $A$ is symmetric matrix, then $A^{\prime }=A$)
Hence, $A{A}^{\prime }$ is symmetric.
So, the given statement is true.