Given two symmetric matrices $A$ and $B$ of the same size (with components $a_{ij}$ and $b_{ij}$ respectively), is it possible to form a larger matrix $C$ with components selected from $\{0,\pm 1,\pm a_{ij}, \pm b_{ij} \}$ such that det(C) = det(A) - det(B) ?
For example, for the boring case of $A$ and $B$ being 1x1 matrices, $$A = \begin{bmatrix} a_{00} \end{bmatrix}, B = \begin{bmatrix} b_{00} \end{bmatrix}, C = \begin{bmatrix} a_{00} & b_{00} \\ 1 & 1 \end{bmatrix}$$ $$\det(C) = a_{00} - b_{00} = \det(A) - \det(B)$$
How can we construct $C$ for larger cases like $A,B$ being 2x2, or 3x3?
Is there a general rule to do this for NxN?
If not, can it be proven to be impossible?