Question

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

Answer 1

We need to prove B'AB is symmetric if A is symmetric and B'AB is skew symmetric if A is skew-symmetric.

(i)Consider A to be a symmetric matrix, then

A'=A (1)

Consider,

( B'AB )'=( AB )'( B' )' =B'A'B        [ ( AB )'=B'A' ] =B'AB         [ from (1) ]

Hence, B'AB is a symmetric matrix.

(ii) Consider A be a skew-symmetric matrix, then

A'=−A (2)

Consider,

( B'AB )'=( AB )'( B' )' =B'A'B =B'( −A )B =−B'AB

Thus, B'AB is a skew-symmetric matrix.

Hence, matrix B'AB is symmetric if A is symmetric or skew symmetric if A is skew symmetric.