Let $A \in \mathbb{R}^{n \times n}$ be a square $n$-dimensional real valued matrix. Let $X \in \mathbb{R}^{a \times n}$ and $Y \in \mathbb{R}^{n \times b}$. Define $M_{n-1}=XA^{n-1}Y$ and $M_{n}=XA^nY$. Is it possible, for proper choices of $X,A,Y$, that there exist indices $i,j$ such that $M_{n-1}(i,j)=0$ but $M_n(i,j)\neq 0$? With (i,j), I mean the entry of the matrix located at row $i$ and column $j$.
Can Cayley-Hamilton's theorem help?
It is certainly possible. As a quick example, take $X = Y = I$ and $$ A = \pmatrix{0&1\\1&0} $$ We note that $M_{1}(1,1) = 0$, but $M_2(1,1) = 1$.