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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