proving some interesting properties of these matrices

Question

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.

Answer 1

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 $X+Y=XY+I$.