Question

Show that the matrix B' 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' A … (1) Consider

Thus, if A is a symmetric matrix, then B'AB  is a symmetric matrix. Now, we suppose that A is a skew-symmetric matrix.

Thus, if A is a skew-symmetric matrix, then B' AB is a skew-symmetric matrix. Hence, if A is a symmetric or skew-symmetric matrix, then B'AB is a symmetric or skewsymmetric matrix accordingly.