What are the minimal conditions for this matrix to be invertible?

Question

Let $A>0$ be a $n \times n$ symmetric positive definite (SPD) and $B$ be an $n \times m$ matrix, then form the matrix $$ C = \left[\begin{array}{cc} A & B \\ B^T & 0\end{array}\right],$$ with $0$ being the $m \times m$ null matrix. What are the conditions in order for $C$ to be invertible? Is it enough for $B$ to have full rank? Or if so, is there anything weaker?

Answer 1

If $C^{-1}$ exists, and if $m=n$, then

$$C^{-1} = \begin{bmatrix}\mathbf{0}& Q^T\\Q& - QAQ^T \end{bmatrix}$$

where $QB=BQ=I$.

So for the case $m=n$, $B$ needs to have an inverse.