# The solution space of $A_{12}X=0$ and $B_{12}X=0$ is isomorphic?

Let $$A$$ be a $$n\times n$$ invertible matrix. Suppose $$A=\begin{pmatrix}A_{11}&A_{12}\\ A_{21}&A_{22} \end{pmatrix}$$ $$A^{-1}=\begin{pmatrix}B_{11}&B_{12}\\ B_{21}&B_{22} \end{pmatrix}.$$ where $$A_{11}$$ is a $$l\times k$$ matrix, $$B_{11}$$ is a $$k\times l$$ matrix, $$1. How to show $$W=\{\alpha; A_{12}\alpha=0\}$$ and $$U=\{\beta; B_{12}\beta=0\}$$ have the the dimension, so is isomorphic.

Intuitively, we have to show $$(n-k)-rank A_{12}=(n-l)-rank(B_{12}).$$ But it sounds impossible at the first glance.