# Singularity of symmetric block matrix with singular diagonal blocks

by Minji Kim   Last Updated September 11, 2019 19:20 PM

I proved that following statement holds true and I think I've seen this somewhere before, but I cannot find the reference that explicitly states about this:

$$\begin{bmatrix}A & B \\ B^T & 0 \end{bmatrix}$$ is invertible if and only if $$C^TAC$$ is invertible where $$A \in \Re^{n \times n}$$ is a symmetric matrix, $$B \in \Re^{n \times p}$$ is a full column rank matrix ($$n>p$$), and $$C \in \Re^{n \times (n-p)}$$ is a full column rank matrix with $$C^TB = 0$$.

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