Integer square $2 \times 2$ block matrix inverse

Question

Let $\mathbf{M}$ be an integer square $2 \times 2$ block matrix $$ \mathbf{M} = \left( \begin{array}{cc} \mathbf{A} & \mathbf{B} \\ \mathbf{C} & \mathbf{D} \end{array} \right) , $$ where $\mathbf{A}$ and $\mathbf{D}$ are square matrices (not necessarily of the same size). Is there a way of testing if $\mathbf{M}$ is regular ($\det (\mathbf{M}) = \pm 1$) in terms of $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$ and $\mathbf{D}$?

For example, is it known some expression of $\det (\mathbf{M})$ in terms of $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$ and $\mathbf{D}$ without any extra assumption (such as some regularity or certain commutativity relations) on the blocks? And if you assume that $\mathbf{A}$ is an $1 \times 1$ matrix (i.e. $\mathbf{A} \in \mathbb{Z}$)?

Answer 1

You can get conditions from the Jacobi identity or (especially in your special case where $\mathbf{A} \in \mathbb{Z}$) from the formulas here. (third equation, in particular)

Answer 2

When $A$ is a $1 \times 1$ matrix:

Let $d = \det D$.

1. if $d \neq 0$, then $\det M = d (A - B D^{-1} C) = dA - B \ adj(D) \ C$, where $adj(D)$ is the adjugate matrix of $D$.
2. since both sides of this equality are continuous functions of the elements of $D$, this equality holds also when $d = 0$, i.e. when $D$ is singular.