Answer to Question #190373 in Linear Algebra for Regomoditswe Dibob

Question

Show that if A is an n × n matrix, then AAT

and A + A

T

are symmetric.

Answer 1

Let A be an $n\times n$ matrix.

1. Let $B=AA^T$

  Take transpose on both sides-

   $B^T=(AA^T)^T\\ B^T=(A^t)^T.A^T\\ B^T=AA^T$

    As, $B^T=B=AA^T$

  So $AA^T$ is symmetric.

2.Let $C=A+A^T$

   Take transpose on both sides-

 $C^T=(A+A^T)^T\\ C^T=A^T+(A^T)^T\\ C^T=A+A^T$

 hence $C=C^T=A+A^T$

So $A+A^T$ is symmetric.