Question

If A is symmetric matrix, then what about B’ AB

Answer 1

A square matrix A=[aij] is said to be symmetric if A'=A that is [aij]=[aji] for all possible value of i and j.

A square matrix A=[aij] is said to be skew symmetric if A'=-A that is [aij]=−[aji] for all possible value of i and j.

Step 1: Let A be symmetric matrix

A'=A

Now consider B'AB and take transpose of it.

(B'AB)'=(B'(AB))'

=(AB)'(B')'

=(B'A')B (From the property of transpose of a matrix we have (AB)'=B'A')

Replace A'=A in the above equation, we get

=B'AB

A matrix is said to be symmetric if A=A'

Thus (B'AB)= B'AB

Hence B'AB is a symmetric matrix

Step 2: Let A be skew symmetric matrix

A'=-A

Now consider B'AB and take transpose of it.

(B'(AB))'=(AB)'B'

=(B'A')B(From the property of transpose of a matrix we have (AB)'=B'A')

Replace A=-A in the above equation we get

=B'(-A)B

=-B'AB

A matrix is said to be skew symmetric if A'= - A

(B'AB) = -B'AB