determinant of a $2\times2$ matrix, sufficiency of inverse

Question

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?

Answer 1

Condition (2) as written is not equivalent to invertibility of a matrix over any (commutative) ring and it should be $$ a_{11}a_{22}\ne a_{12}a_{21} $$ A truly equivalent condition is that the determinant is invertible in the ring, so it shouldn't be really difficult to find an example over the simplest ring which is not a field.

Answer 2

Try $\begin{pmatrix}1 & 1\\ 0 & 2\end{pmatrix}\in M_2(\mathbb{Z}_4)$. (2) doesn't imply (1) for the reason that $2$ is no invertible. Just as mentioned before by @egreg

Answer 3

Hint:

for $$ a_{11}a_{22}\ne a_{12}a_{21} $$ consider the subring of $M_2(\mathbb{R})$ with matrices of the form: $$ \begin{bmatrix} \mathbb{Z}&\mathbb{Q}\\ 0&\mathbb{Z} \end{bmatrix} $$

Many matrices with non null determinant are not invertible in this ring.

Answer 4

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 $M_2(\Bbb R)$.