Let $B$ and $C$ be two matrices of $\mathbf{GL}_n$. Let $A \in \mathbf{M}_{2n}$ be the matrix defined by blocks as follows:
$$A = \left(\begin{array}{cc} B & 0_n \\ 0_n & C \end{array}\right)$$
Deduce that if $C$ is similar to $B^{-1}$, then $A$ is a product of two symmetry matrices.