What are necessary and sufficient conditions such that $X^2+Y^2$ is invertible?

Question

Let $X,Y$ be $n\times n$ matrices such that $X^3=Y^3$ and $X^2Y=Y^2X$. What are necessary and sufficient conditions on $X$ and $Y$ such that $X^2+Y^2$ is invertible?

I believe $X=Y$ is a sufficient condition, but how does one find all the necessary and sufficient conditions?

I tried writing it out individually as $XXX=YYY$ etc, but just found jumbles of $X, Y, X^{-1}$, and $Y^{-1}$ which really don't help me much. The problem is that as far as I know there is no simple way to link $\det(A+B)$ to $\det A$ and $\det B$.

However, I am not looking for a full answer, but rather for hints... I already have the answer but am choosing not to look.

Answer 1

To help you get started, note that $(X^2+Y^2)(X-Y) = X^3-Y^3+Y^2X-X^2Y = 0$. Mouseover the lines below if you need more details.

So if $X^2+Y^2$ is invertible, then we must have $X-Y = (X^2+Y^2)^{-1}0 = 0$, i.e. $X = Y$.

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It is easy to show that if $X = Y$ and $X$ is invertible, then $X^2+Y^2 = 2X^2$ is invertible, and if $X = Y$ but $X$ is not invertible, then $X^2+Y^2 = 2X^2$ is not invertible.

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Therefore, $X^2+Y^2$ is invertible iff $X = Y$ and $X$ is invertible.