Question

Let A and B be two nonsingular square matrices, A${}^T$ and B${}^T$ are the transpose matrices A and B, respectively, then which of the following are correct?

• A. B${}^T$AB is symmetric matrix if A is symmetric
• B. B${}^T$AB is symmetric matrix if B is symmetric
• C. B${}^T$AB is skew-symmetric matrix for every matrix A
• D. B${}^T$AB is skew-symmetric matrix if A is skew-symmetric

Answer 1

Answer: (A), (D)

The correct options are
A B${}^T$AB is symmetric matrix if A is symmetric
D B${}^T$AB is skew-symmetric matrix if A is skew-symmetric

To check whether $B^{T}AB$ is a symmetric matrix.

Applying transpose,we obtain

$(B^{T}AB{)}^T$

$\Rightarrow (AB{)}^T.(B^T{)}^T$

$\Rightarrow B^T{A}^{T}B$

In the above result if $A^T=A$, then $B^{T}AB$ becomes a symmetric matrix.

Therefore, $B^{T}AB$ is symmteric matrix if $A$ is symmetric.

In the above result if $A^T=-A$ then $B^{T}AB$ becomes a skew-symmetric matrix.

Therefore, $B^{T}AB$ is skew-symmetric matrix if $A$ is skew-symmetric.

Hence, the correct options are $\left(A\right)$ and $\left(D\right)$.