Question

Show that matrix $A+B$ is symmetric or skew symmetric according as $A$ and $B$ are symmetric of skew symmetric.

Answer 1

$(A+B{)}^{\prime }=A^{\prime }+B^{\prime }=A+B$ if both $A$ and $B$ are symmetric
$\therefore$ $A+B$ is symmetric
$(A+B{)}^{\prime }=A^{\prime }+B^{\prime }=-A-B=-(A+B)$ if both $A$ and $B$ are skew symmetric
$\therefore$ $A+B$ is skew symmetric.