Determinants of Partitioned Matrices

Theorem

Let

• $A$ : Square matrix
• $A$ partitioned as $$ A=\begin{bmatrix} T & U \ V & W \end{bmatrix}

If $T$ is square and [[3 Reference/def-inverse-matrix_202509241225\|nonsingular]] (invertible) Then

|A|=|T|; |W - VT^{-1}U|

That is, the determinant is the product of the determinant of the [[3 Reference/def-principal-submatrix_202509241406\|principal submatrix]] and the determinants of its Schur component ## Proof

\begin{bmatrix} A_{11} & A_{12} \ A_{21} & A_{22} \end{bmatrix}

\begin{bmatrix} A_{11} & 0 \ A_{21} & A_{22} - A_{21}A_{11}^{-1}A_{12} \end{bmatrix} \begin{bmatrix} I & A_{11}^{-1}A_{12} \ 0 & I \end{bmatrix}