Consider the following formula for the determinant when $A$ is nonsingular, $$ \det \begin{bmatrix} A & B\\ C & D \end{bmatrix} = \det (A) \det (D - C A^{-1}B). $$
I am thinking if $A$ is non-singular, however, the full matrix is singular, then does the following equation hold? $$ Det \begin{bmatrix} A & B\\ C & D \end{bmatrix} = Det (A) Det (D - C A^{-1}B), $$ where $Det$ denotes the pseudo determinant.