每週問題 May 22, 2017

以伴隨矩陣的行列式表達分塊矩陣的行列式。

Suppose A is n\times n, B is n\times 1, C is 1\times n, and d is a number. Prove that

\begin{vmatrix} A&B\\ C&d \end{vmatrix}=d|A|-C(\hbox{adj}A)B.

 
參考解答:

A 是可逆的,使用分塊矩陣的行列式公式,

\begin{aligned} \begin{vmatrix} A&B\\ C&d \end{vmatrix}&=|A|\cdot (d-CA^{-1}B)\\ &=d|A|-C(|A|A^{-1})B\\ &=d|A|-C(\hbox{adj}A)B.\end{aligned}

A 是不可逆的,利用連續論證法來證明。根據矩陣的特徵值性質,存在一個正數 \delta 使得對於 0<\epsilon<\deltaA+\epsilon I 是可逆矩陣。套用前面結果,

\displaystyle  \begin{vmatrix}  A+\epsilon I&B\\  C&d  \end{vmatrix}=d|A+\epsilon I|-C(\hbox{adj}(A+\epsilon I))B

\epsilon\to 0,即證得所求。

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