Prove $ X = \left(\begin{array} &A & C \\ 0 & B \end{array} \right) $ is invertible iff A and B both are.

Question

Suppose $A$ is a $n \times n$ matrix, $B$ is a $m \times m$ matrix, and $C$ is a $n \times m$ matrix. Prove $ X = \left(\begin{array} &A & C \\ 0 & B \end{array} \right) $ is invertible iff A and B are.

Answer 1

$(1)$ Suppose $A,B$ are both invertible. What is the most obvious candidate for an inverse of your matrix?

$(2)$ Suppose your matrix is invertible. Write its inverse in blocks as your original matrix. What is the obvious choice for inverses of $A$ and $B$?

ADD A slicker solution would be to note $$\det \begin{pmatrix} A & C \\ {0} & B \end{pmatrix}=\det A\cdot \det B $$

Answer 2

Your matrix has full rank if and only if both $A$ and $B$ do.

OR

The columns of your matrix are linearly independent if and only if the columns of $A$ and $B$ are.