If $X^2 = YX$ for some non-singular matrix $Y$, then $\operatorname{Null}(X) \cap \operatorname{Range}(X) = \{ 0\}$

Question

There are two square matrices $X,Y$ of size $m \times m$. If $Y$ is non-singular matrix and $X^2 =YX$, prove that: $$\operatorname{Null}(X) \cap \operatorname{Range}(X) = \{ 0\}$$

My try:

Let $\{x_1, x_2,...,x_r \}$ is basis of $\operatorname{Null}(X)$.

Let $\{y_1, y_2,...,y_t \}$ is basis of $\operatorname{Range}(X)$.

I hope that to prove the two bases are independent, but I get stuck.

Answer 1

Take $z\in\text{Null}(X)\cap\text{Range}(X)$. Need to show that $z=0$.

1. $z\in\text{Null}(X)$ $\Leftrightarrow$ $Xz=0$.
2. $z\in\text{Range}(X)$ $\Leftrightarrow$ $\exists w\colon$ $z=Xw$.

"1+2" gives $X^2w=0$ $\Leftrightarrow$ $YXw=0$ $\Leftrightarrow$ $z=Y^{-1}YXw=0$.