rank of certain block matrix need to clarified

Question

Given that $E$ is an $n_1\times n_1$ matrix and $C$ is a $p_1\times n_1$ matrix. We are given that rank of $\begin{bmatrix}I_{n_1}-E\\C\end{bmatrix}=rank(C)$ then we need to show

rank of $\begin{bmatrix}E\\C\end{bmatrix}=n_1$

Thanks for helping.

Answer 1

The rank hypothesis amounts to saying that $$CX = 0\quad \Longrightarrow\quad E X = X$$ For any $X\in {\mathbb K}^{n_1}$, suppose that $$\binom{E}{C}X = 0$$ it follows that $CX = 0$ and $E X = 0$, but then $X = EX=0$. The nullspace of $\binom{E}{C}$ being $\{0\}$, it follows that its rank is $n_1$.