Answer to Question #200935 in Linear Algebra for andisiwe

Question

prove that A is a square matrix, then AA^T and A + A^T are symmetric

Answer 1

A square matrix whose transpose is equal to itself is called a symmetric matrix; that is, B is symmetric if $B^T=B,B$ is a square matrix.

Transpose of a product $(CD)^T=D^TC^T$

The operation of taking the transpose is an involution (self-inverse): $(B^T)^T=B$

The transpose respects addition: $(C+D)^T=C^T+D^T$

Then

$$(AA^T)^T=(A^T)^TA^T=AA^T$$

$AA^T$ is symmetric.

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

$A+A^T$ is symmetric.