I could do this problem with bruteforce but I think there must be some elegant theorem that helps to calculate the determinant with the block matrix (here having symmetric matrices inside) such as:
$B=\begin{pmatrix}1 & 1 & 4 & 5 \\ 1 & 1 & 5 & 4 \\ 2 & 4 & 1 & 1 \\ 4 & 2 & 1 & 1 \\\ \end{pmatrix}=\begin{pmatrix}I_{2,2} & S_{2,2,1} \\ S_{2,2,2} & I_{2,2}\end{pmatrix}$
Actually, look this one
$B= \begin{pmatrix}1 & 1 & 2 & 4 \\ 1 & 1 & 4 & 2 \\ 2 & 4 & 1 & 1 \\ 4 & 2 & 1 & 1 \\ \end{pmatrix}+ \begin{pmatrix} 0 & 0 & 2 & 1 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\\ \end{pmatrix}$
and now I am thinking how I could use this one to speed up the calculation...determinant over this-kind-of-matrix-sum?
The problem is booringly stated as with Gaus method but I am interested to find some trick to calculate the determinante. My first idea was to do $4-3$ -row-minus and $1-2$ -row-minus (so getting some ones away but there must be some theorem to simplify the monotonous Gaussian elimination and determinant finding).
Page 741 here.
References by J.D. for further research