how to show this has an eigenvector

Question

I am given that rank of $K=[B, AB,A^2B,\dots, A^{n-1}B]<n$ where $A$ is a $n\times n$ and $B$ is $n\times m$ matrix. I need to show that there exists a row-vector $x \in \Bbb C^n$ such that

$xA=\lambda x, xB=0$

Answer 1

Hint: Observe that the left null space of $[B, AB,\dots, A^{n-1}B]$ is invariant under right multiplication by $A$.