Question

Prove that if b is a square matrix then b+b^t is symmetric

Answer 1

A matrix $\mathbf X$ is symmetric if $\mathbf X=\mathbf X^\top$. We have

$(\mathbf B+\mathbf B^\top)^\top=\mathbf B^\top+(\mathbf B^\top)^\top=\mathbf B^\top+\mathbf B=\mathbf B+\mathbf B^\top$

and so $\mathbf B+\mathbf B^\top$ is indeed symmetric.