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:
$$A = \left(\begin{array}{cc} B & 0_n \\ 0_n & C \end{array}\right)$$
Let $S_1$ be the block matrix
$$S_1 = \left(\begin{array}{cc} 0_n & P \\ Q & 0_n \end{array}\right),$$
where $P, Q$ are two elements of $\mathbf{GL}_n$.\\
Determine the conditions relating $B, C, P, Q$ for the matrices $S_1$ and $S_2 = S_1 A$ to be symmetry matrices.