Let $\times$ denote the (associative) operation $a \times b = a + b + ab$ (with $+$ defined as usual) and consider the ring $(\mathbb{Z}, \times, +)$.
Find the inverse of the $2 \times 2$ matrix:
$$ \left[ \begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array} \right]^{-1} = \;\;? $$
i have that $(\mathbb{Z}, \times)$ is a group with $a \times b = a + b + ab$. perhaps there's no operation that distributes over this
We're looking for the inverse of the matrix $A = \left[\begin{smallmatrix}1&2\\3&4\end{smallmatrix}\right]$, with respect to the operation $\times$ that you have defined.
Hint: Verify that the $0$-matrix is the multiplicative identity, i.e. $a \times 0 = 0 \times a = a$ for all matrices $a$. With that in mind, it suffices to solve $A \times X = 0$, i.e. $$ A + X + AX = 0 $$ for $X$.