<s>[INST] Can I use this formula with pseudo determinants instead of usual determinants? Let $A$ be a matrix with $A^+$ Moore-Penrose inverse. Let also $Det()$ denote the pseudo-determinant of a matrix.
Does the formula (which assumes the existence of $A^{-1}$)
$$ det\left( \begin{array}{cc}
A & B  \\
C & D   \end{array} \right) = det(A)det(D-CA^{-1}B), $$
where $det()$ denotes the usual determinant, applies to the use of pseudo determinants and Moore-Penrose inverse? This is
$$ Det\left( \begin{array}{cc}
A & B  \\
C & D   \end{array} \right) = Det(A)Det(D-CA^{+}B)\,? $$
 [/INST] The classical formula
${\rm Det}(R) = {\rm Det}( \left[ \begin{array}{cc} A & B \\ C & D \end{array} \right] )
= {\rm Det}(D) {\rm Det}(A-BD^+C)$ for block matrices
does not hold for pseudo determinants ${\rm Det}$ and Moore pseudo inverse $D^+$, as already
$$ \left[ \begin{array}{cc} A & B \\ C & D \end{array} \right] 
   \left[ \begin{array}{cc} I & 0 \\ -D^+ C & I \end{array} \right] 
 = \left[ \begin{array}{cc} A-B D^+ C & B \\ 0 & D \end{array} \right] $$
does not holds with the pseudo inverse $D^+$. But here is an example:
$$ A=\left[ \begin{array}{cc} -1 & 0 \\ -2 & -1 \\ \end{array} \right],
   B=\left[ \begin{array}{cc} 2 & 1 \\ 0 & 2 \\ \end{array} \right], 
   C=\left[ \begin{array}{cc} 2 & -2 \\ 2 & -1 \\ \end{array} \right], 
   D=\left[ \begin{array}{cc} 1 & 1 \\ 0 & 0 \\ \end{array} \right] $$
so that the block matrix is
$$ R=\left[ \begin{array}{cccc} -1 & 0 & 2 & 1 \\ -2 & -1 & 0 & 2 \\ 2 & -2 & 1 & 1 \\
                                 2 & -1 & 0 & 0 \\ \end{array} \right] $$
is even invertible with $\det(R)={\rm Det}(R)=2$. The pseudo inverse of
$D$ is $D^+ = \left[ \begin{array}{cc} \frac{1}{2} & 0 \\ \frac{1}{2} & 0 \\ \end{array} \right]$.
Also the matrix $A - B D^+ C= \left[ \begin{array}{cc} -4 & 3 \\ -4 & 1 \\ \end{array} \right]$
is invertible with determinant $8$. We have ${\rm Det}(D)=1$. The left hand side is $2$,
the right hand side is $8$. It is a nice question as we can answer also the question whether the block diagonal formula for diagonal block matrices generalizes to upper triangular block matrices.
There are a few surprises with pseudo determinants: see my
(paper) about it and the
(ArXiv pre-print).
</s>