Question

Let A and B are two non-singular square matrices, $A^T$ and $B^T$ are the transpose matrices of A and B respectively, then which of the following is correct

• A. $B^{T}AB$ is symmetric matrix if and only if A is symmetric
• B. $B^{T}AB$ is symmetric matrix if and only 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 B is skew symmetric

Answer 1

The correct option is A $B^{T}AB$ is symmetric matrix if and only if A is symmetric

We are asked about whether the expression $(B^{T}AB{)}^T$ is symmetric or skew symmetric uncer various conditions.
So let us start by considering the transpose of the expression.
$(B^{T}AB{)}^T=B^T{A}^T(B^T{)}^T=B^T{A}^{T}B$
i.e, $(B^{T}AB{)}^T=B^T{A}^{T}B$------------------(1)
case 1: A is symmetric .
$(B^{T}AB{)}^T=B^{T}AB$
This implies that the given expression is symmetric
case 2: B is skew-symmetric .
$(B^{T}AB{)}^T=B^T{A}^{T}B=(-B)A^{T}B\ne -B^{T}AB$
$\therefore B^{T}AB$ is not skew symmetric if B is skew symmetric
You can similarly verify that the other two cases are doesn't lead to a right option.