Nonsingularity of a block matrix

Question

Let $X=\left(\begin{array}{cc} A & B\\ C & 0 \end{array}\right)$ and:

If $X$ is non-singular, is $A$ non-singular when $B$ is full column rank and $C$ is full row rank?

Answer 1

Counterexample:

$$\begin{pmatrix}0&1\\1&0\end{pmatrix}$$