Question

Prove that the matrix $B^{\prime }AB$ is symmetric or skew symmetric according as $A$ is symmetric or skew symmetric.

Answer 1

If $A$ be symmetric i.e., $A^{\prime }=A$ then
$(B^{\prime }AB{)}^{\prime }=[B^{\prime }\left(AB\right){]}^{\prime }=\left(AB{)}^{\prime }\right(B^{\prime }{)}^{\prime }$
$=\left(B^{\prime }{A}^{\prime }\right)B=B^{\prime }{A}^{\prime }B=B^{\prime }AB$
Hence $B^{\prime }AB$ is symmetric
If $A$ is skew symmetric i.e., $A^{\prime }=-A$ then
$(B^{\prime }AB{)}^{\prime }=[B^{\prime }\left(AB\right){]}^{\prime }=\left(AB{)}^{\prime }\right(B^{\prime }{)}^{\prime }$
$=\left(B^{\prime }{A}^{\prime }\right)B=B^{\prime }(-A)B=-\left(B^{\prime }AB\right)$
$\therefore$ $B^{\prime }AB$ is skew symmetric