Find two distinct, non-zero matrices, A and B, such that A*B+A+B = 0

Question

this is a problem which is for homework in my math course. The problem states that you must find two distinct, non-zero matrices, (Size 2x2) such that A * B + A + B = 0.

I'm not really looking for an answer, but rather the methodology I should be using to come to this answer. It seems like the easiest way to do this would be through brute force, but I am somewhat slow when it comes to math and so I was hoping there might be an easier method out there. Thank you in advance.

Answer 1

Take any number $a\ne-1$. Then take $b=-\frac a{a+1}$. With this choice, $ab+a+b=0$. Now, take $A=\left[\begin{smallmatrix}a&0\\0&a\end{smallmatrix}\right]$ and $B=\left[\begin{smallmatrix}b&0\\0&b\end{smallmatrix}\right]$.

Answer 2

$AB+A+B=0$ is equivalent to $(A+I)(B+I)=AB+A+B+I=I$ for the identity matrix $I=\begin{bmatrix}1 &0\\0&1\end{bmatrix}$.

Thus, picking any invertible matrix $M$, the matrices $A=M-I$ and $B=M^{-1}-I$ are solutions. In fact, it is easy to show that these are the only solutions.