Matrices rank problem

Question

$X\in \text{Mat}_n (\mathbb{R} )$ and $|X|\neq 0$. $X$ has column vectors $X_1,X_2,\ldots ,X_n$. $Y$ is a matrix that consists of column vectors $X_2,X_3,\ldots ,X_n,0$. Let $A=YX^{-1}$ and $B=X^{-1}Y$. Find rank $A$ and rank $B$.

It's clear from $|AB|=|A||B|$ that $A$ and $B$ rank must be $<n$. How could I get definite answer here?

Answer 1

Hint: Use $\text{rank}\left(MN\right)\leq \min\left(\text{rank}(M),\text{rank}(N)\right)$.

Answer: Since $X$ is invertible, $\text{rank}\left(X\right)=\text{rank}\left(X^{-1}\right)=n$. On the one hand $\text{rank}\left(A\right)=\text{rank}\left(YX^{-1}\right)\leq \min\left(\text{rank}\left(Y\right), \text{rank}\left(X^{-1}\right)\right)=\text{rank}\left(Y\right)$. And on the other, since $Y=AX$, similarly follows that $\text{rank}\left(Y\right)\leq \text{rank}\left(A\right)$. So $\text{rank}\left(A\right)=\text{rank}\left(Y\right)=n-1.$