Question

Prove that if A is a square matrix then AA^T and A + A are symmetric

Answer 1

Let A be a square matrix.

Consider

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

Hence $AA^T$ is symmetric

Also, consider

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

Hence $A+A^T$ is symmetric.