I want to solve the matrix equation for $X$
$$aX^{3} + bX^{2} = I,$$ where $a,b \in \mathbb{R}$ and $X \in \mathbb{R}^{n\times n}$.
My thoughts:
If $a = 0$ or $b = 0$, the solution is easy.
If $a, b \neq 0$ I have tried to find matrices $U$ and $V$ such that $X^3 =UD_3V$ and $X^2 =UD_2V$, where $D_2$ and $D_3$ are diagonal matrices, but I did not have success until now. Maybe this is not possible.
Do you have any idea?
Thanks in advance!