Answer to Question #194045 in Linear Algebra for Mpopo

Question

Prove that if A is a square matrix then AAT and A + A T are symmetric.

Answer 1

Let A be any matrix.

Also let B=AA^T

Now $B^T =(AA^T )^T =(A^T )^T A^T =AA^T$

=B

[ Since (A^T)^T=A]

So AA^T

is a symmetric matrix.

Let B=A+A^T

Now, B^T=A^T+A [ Since (A^T)^T=A]

or, B^T=B.

So B is symmetric matrix.

And B= A+A^T