determinant of a $2\times2$ matrix, sufficiency of inverse What is the simplest example of a nontrivial ring in which these two conditions are not equivalent for any $2\times 2$ matrix $A$:
(1) there is a $2\times2$ matrix $B$ with $A\cdot B=1$
(2) $a_{11}a_{22}\neq a_{21}a_{21}$
To make the question more interesting: can the structures such that (1) is (not) equivalent to (2) be characterized?

A matrix $A\in M_2(R)$, where $R$ is a ring, is invertible iff $\det A\in U(R)$, where $U(R)$ is the set of the units of the ring $R$.
So a simple example is given by the ring $\Bbb Z$, any matrices whose determinant is different to $\pm 1$ is not invertible.

Example 
  $$ A=
\begin{pmatrix}
2 & 1\\
3 & 4
\end{pmatrix}
$$
$\det A=5$ so $A$ is not invertible in $M_2(\Bbb Z)$, even if is invertible in $M_2(\Bbb R)$. 

