Question

Which of the following is correct?

विकल्प

• B' AB is symmetric if A is symmetric.

• B' AB is skew-symmetric if A is symmetric.

• B' AB is symmetric if A is skew-symmetric.

• B' AB is symmetric if A is skew-symmetric.

Answer 1

B' AB is symmetric if A is symmetric.

Explanation:

Let A be a symmetric matrix.

Then A' = A

Now, (B'AB)' = B'A'(B')'    [∵ (AB)' = B'A']

= B'A'B    [∵ (B)' = B]

= B'AB    [∵ A' = A]

⇒ B'AB is a symmetric matrix.

Now, let A be a skew-symmetric matrix.

Then A' = −A

∴ (B'AB)' = B'A'(В')'    [∵ (AB)' = B'A']

=B'A'B   [∵ (B')' = B]

= −B'AВ

∴ B'AB is a skew-symmetric matrix.