## 每週問題 June 10, 2013

Let $A, B, C, D$ be $n\times n$ matrices. Show that if $A$ is nonsingular, then

$\begin{vmatrix} A&B\\ C&D \end{vmatrix}=(\det A)(\det (D-CA^{-1}B)).$

$A$ 可逆，考慮下列分塊乘法，

$\begin{bmatrix} I&0\\ -CA^{-1}&I \end{bmatrix}\begin{bmatrix} A&B\\ C&D \end{bmatrix}=\begin{bmatrix} A&B\\ 0&D-CA^{-1}B \end{bmatrix}$

$\begin{vmatrix} I&0\\ -CA^{-1}&I \end{vmatrix}\cdot\begin{vmatrix} A&B\\ C&D \end{vmatrix}=\begin{vmatrix} A&B\\ 0&D-CA^{-1}B \end{vmatrix}$

$\begin{vmatrix} A&B\\ C&D \end{vmatrix}=\begin{vmatrix} A \end{vmatrix}\cdot\begin{vmatrix} D-CA^{-1}B \end{vmatrix}$

PowSol-June-10-13