First slide

Transpose of matrix

Question

Which of the following is correct?

Moderate

A

B′AB is symmetric if A is symmetric
B

B′AB is skew-symmetric if A is symmetric
C

B′AB is symmetric if A is skew-symmetric
D

B′AB is skew-symmetric if A is skew-symmetric

Solution

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′(B′)′ [ $∵$ (AB)′ = B′A′]
= B′A′B [ $∵$ (B′)′ = B]
= B′(–A)B [ $∵$ A′ = –A]
= –B′AB
$∴$ B′AB is a skew-symmetric matrix.

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