I had a doubt, can we apply the determinant of a 2x2 Matrix to: $$\begin{vmatrix} E_{m} & A \\ B & E_{n} \\ \end{vmatrix} $$ with $A \in K^{m\times n}$ and $B \in K^{n\times m} $ and $E$ as Identity matrix of respectively $K^{m\times m}, K^{n\times n} $ so that it is $E_{m}E_{n}-AB$, also when the matrix product for the $E$s is not defined?
2026-03-27 00:09:35.1774570175
Determinants of Block Matrix
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$$\pmatrix{E&A\\B&E}=\pmatrix{E-AB&A\\O&E}\pmatrix{E&O\\B&E}$$ etc.