. Show that the matrix B’AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.

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Case1: Given that A is symmetric. We will prove $B’AB$ is symmetric. As A is symmetric, so $A' = A$.

Now, $(B'AB)' = B'A'(B')' = B'A'B = B'AB$
Thus, $B'AB$ is a symmetric matrix.

Case II: Given is skew symmetric, i.e.,
$A' = -$A. We will prove that $B'AB$ is skew symmetric.

Now, $(B'AB)' = B'A'(B')' = B'A'B$

=$B'($ - $A)B=$ - $B'AB$

Hence, $B'AB$ is a skew-symmetric matrix.