Question

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

Answer 1

We suppose that $A$ is a symmetric matrix, then $A^{\prime }=A$.........(1)

Consider ${\left(B^{\prime }AB\right)}^{\prime }={\left\{B^{\prime }\left(AB\right)\right\}}^{\prime }$

$={\left(AB\right)}^{\prime }{\left(B^{\prime }\right)}^{\prime }[{\left(AB\right)}^{\prime }=B^{\prime }{A}^{\prime }]$

$=B^{\prime }{A}^{\prime }\left(B\right)[{\left(B^{\prime }\right)}^{\prime }=B]$

$=B^{\prime }\left(A^{\prime }B\right)$

$=B^{\prime }\left(AB\right)$ [using (1)]

$\therefore {\left(B^{\prime }AB\right)}^{\prime }=B^{\prime }\left(AB\right)$

Thus, if $A$ is a symmetric matrix, then $B^{\prime }AB$ is a symmetric matrix.

Now, we suppose that $A$ is a skew-symmetric matrix.

Then, $A^{\prime }=-A$

Consider

${\left(B^{\prime }AB\right)}^{\prime }={\left[B^{\prime }\left(AB\right)\right]}^{\prime }={\left(AB\right)}^{\prime }{\left(B^{\prime }\right)}^{\prime }$

$=\left(B^{\prime }{A}^{\prime }\right)B=B^{\prime }(-A)B$

$=-B^{\prime }AB$

$\therefore {\left(B^{\prime }AB\right)}^{\prime }=-B^{\prime }AB$

Thus, if $A$ is a skew-symmetric matrix, then $B^{\prime }AB$ is a skew-symmetric matrix.

Hence, if $A$ is a symmetric or skew-symmetric matrix , then $B^{\prime }AB$ is a symmetric or skew-symmetric matrix accordingly.