If the Rank of [A - B], where A and B are m x m matrix, is not full then there exist a matrix $\alpha$ such that $\alpha$ [A-B ] = 0. Why?

Question

I am trying to solve a singular linear system, where

$A y_{t+1} = B y_{t} + C x_t$ where y (m x 1) is endogenous variables and x(n x 1) are exogenous variables. The condition for solution to exist is [Az-B] $\neq$ 0 identically in $z$. Image gives the logic behind putting such a condition. I want to understand the reason why the $\alpha$ exists.