show that the matrix B'AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.?

Case I
when A is symmetric, i.e, A'=A
                     (B'AB)' = (AB)'(B')'
                                 =B'A'B
            but because A'=A
            this implies,  (B'AB)'=B'AB
​           i.e, it is symmetric.
Case II
when A is skew symmetric, i.e, A'=-A
                      (B'AB)'​= (AB)'(B')'
                                   =B'A'B​
            but A'=-A
            this implies,  (B'AB)'​=-B'AB
           i.e, it is skew symmetric.
Hence, matrix  B'AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.