<s>[INST] proving some interesting properties of these matrices let X and Y be two matrices different from I, such that $XY=YX$ and $X^n-Y^n$ is invertible for some natural number n .If
$$X^n-Y^n = X^{n+1}-Y^{n+1} = X^{n+2}-Y^{n+2}$$,
then prove that $I-X,I-Y $ are singular and $X+Y=XY+I$
my approach:
I tried to pre multiply and post multiply by $X and Y$ but it could not produce anything.Kindly help me by providing some suggestions on how to solve this question.
 [/INST] From $X^n-Y^n=X^{n+1}-Y^{n+1}=X^{n+2}-Y^{n+2}$, we obtain
\begin{align}
X^n(I-X) &= Y^n(I-Y),\tag{1}\\
X^{n+1}(I-X) &= Y^{n+1}(I-Y).\tag{2}
\end{align}
Subtract both sides of $(1)$ by $X^n(I-Y)$, we obtain
\begin{align}
(Y-X)X^n &= (Y^n-X^n)(I-Y).\tag{3}
\end{align}
Substitute the LHS of (1) into the RHS of (2), we get
\begin{align}
X^{n+1}(I-X) &= YX^n(I-X),\\
(X-Y)X^n(I-X) &= 0,\\
(X^n-Y^n)(I-Y)(I-X) &= 0\ \text{ by $(3)$},\\
(I-Y)(I-X) &= 0. \tag{4}
\end{align}
Since $X,Y\ne I$, $(4)$ implies that neither $I-Y$ nor $I-X$ is invertible. Also, by expanding $(4)$, we obtain $X+Y=XY+I$.
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