## 每週問題 October 5, 2015

Let $A=\begin{bmatrix} B&C\\ D&E \end{bmatrix}$ be an $n\times n$ nonsingular matrix, where $B$ is $k\times k$. Denote the adjugate of $A$ by $\hbox{adj}\,A=\begin{bmatrix} P&Q\\ R&S \end{bmatrix}$, where $P$ is $k\times k$. Prove the Jacobi identity

$\det P=(\det A)^{k-1}(\det E)$.

$\begin{bmatrix} P&Q\\ R&S \end{bmatrix}\begin{bmatrix} B&C\\ D&E \end{bmatrix}=\begin{bmatrix} (\det A)I_k&0\\ 0&(\det A)I_{n-k} \end{bmatrix}$

$\begin{bmatrix} P&Q\\ 0&I_{n-k} \end{bmatrix}A=\begin{bmatrix} P&Q\\ 0&I_{n-k} \end{bmatrix}\begin{bmatrix} B&C\\ D&E \end{bmatrix}=\begin{bmatrix} (\det A)I_k&0\\ D&E \end{bmatrix}$

$(\det P)(\det A)=\left(\det((\det A)I_k)\right)(\det E)=(\det A)^k(\det E)$