Let $A$ and $B$ be $2$ by $2$ matrices such that $A^2$ $+$ $A$ $=$ $0$ and $B^2$ $+$ $B$ $=$ $0$. What is an example of $A$ and $B$ such that $(A + B)^2$ $+$ $A$ $+$ $B$ does not equal $0$?
I am struggling to find matrices that meet the initial conditions other than the $0$ matrix. Is there a general formula through which this could be satisfied? Moreover, how would we move on to the main part after that?
Please note that this is only a sub part of a larger problem. But I need to figure this out first.
Any help?
$A=B=-I$ is one example, and it works for all sizes. Since it annihilates a square-free polynomial, a matrix satisfies the initial condition if and only if it is diagonalizable in $\Bbb C$ and its eigenvalues are all roots of that polynomial (in this case, roots of $x^2+x$).