If B is a symmetric matrix,

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Question:

If $B$ is a symmetric matrix, write whether the matrix $A B A^{\top}$ is symmetric or skew-symmetric.

Solution:

If $B$ is a symmetric matrix, then $B^{T}=B$.

$\left(A B A^{T}\right)^{T}=\left(A^{T}\right)^{T} B^{T} A^{T} \quad\left[\because(A B C)^{T}=C^{T} B^{T} A^{T}\right]$

$\Rightarrow\left(A B A^{T}\right)^{T}=A B^{T} A^{T} \quad\left[\because\left(A^{T}\right)^{T}=A\right]$

$\Rightarrow\left(A B A^{T}\right)^{T}=A B A^{T} \quad\left[\because B^{T}=B\right]$

$\therefore A B A^{T}$ is a symmetric matrix.

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