Let $X$ ,$Y$ be two $n\times n$ real matrices such that $XY=X^2+X+I$.
Which of the following statements are necessarily true?
1.$X$ is invertible.
2.$X+I$ is invertible.
3.$XY=YX$.
4.$Y$ is invertible.
Since $XY=X^2+X+I\implies X(Y-X-I)=I$.Therefore $X$ is invertible.
Now as $X$ is invertible therefore $Y=X+X^{-1}+I$.
So, $YX=(X+X^{-1}+I)X=X^2+X+I=XY$.
Thus option (1) and (3) are correct.
For option(2) I have a matrix $$A=\begin{bmatrix}
-1&0&\\
0&-1
\end{bmatrix}_{2\times 2}$$ Which satisfies the given condition but then $X+I$ is a zero matrix.So option (2) is incorrect.
I am unable to solve option(4).I tried to show that $Y$ is not invertible always by trying counterexamples but did not find any counterexample.(may be $Y$ is invertible).
Best Answer
Consider $X=\left(\begin{matrix}-1&1\\-1&0\end{matrix}\right)$. It's easy to verify that $XY=X^2+X+I=0$. Since $X$ is invertible, then $Y$ cannot be invertible.