张祖锦常用结论08偶数阶方阵行列式的迅速降阶法

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张祖锦常用结论偶数阶方阵行列式的迅速降阶法

设 $\displaystyle A,B,C,D$ 是 $\displaystyle n$ 阶矩阵且 $\displaystyle AC=CA$, 求证:

$$\begin{aligned} \left|\begin{array}{cccccccccc}A&B\\\\ C&D\end{array}\right|=|AD-CB|. \tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\\\mbox{zhangzujin.cn}\end{array}}\end{aligned}$$

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(1)、 若 $\displaystyle A$ 可逆, 则由

$$\begin{aligned} \left(\begin{array}{cccccccccc}A&B\\\\ C&D\end{array}\right)\left(\begin{array}{cccccccccc}E&-A^{-1}B\\\\ 0&E\end{array}\right)=\left(\begin{array}{cccccccccc}A&0\\\\ C&D-CA^{-1}B\end{array}\right) \tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\\\mbox{zhangzujin.cn}\end{array}}\end{aligned}$$

知

$$\begin{aligned} \left|\begin{array}{cccccccccc}A&B\\\\ C&D\end{array}\right|&=|A|\cdot |D-CA^{-1}B| =|AD-AC\cdot A^{-1}B|\\\\ &=|AD-CA\cdot A^{-1}B| =|AD-CB|. \tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\\\mbox{zhangzujin.cn}\end{array}}\end{aligned}$$

(2)、 当 $\displaystyle A$ 不可逆时, 设 $\displaystyle A$ 的非零特征值为 $\displaystyle \lambda_1,\cdots,\lambda_s$, 则对 (为啥交上有理数集, 因为所有数域包含有理数域)

$$\begin{aligned} \forall\ \varepsilon\in \left(0,\min_{1\leq i\leq s}|\lambda_i|\right)\cap \mathbb{Q}, \tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\\\mbox{zhangzujin.cn}\end{array}}\end{aligned}$$

$\displaystyle A^\varepsilon=A+\varepsilon E$ 可逆, 而由第 1 步知

$$\begin{aligned} \left|\begin{array}{cccccccccc}A^\varepsilon&B\\\\ C&D\end{array}\right|=|A^\varepsilon D-CB|. \tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\\\mbox{zhangzujin.cn}\end{array}}\end{aligned}$$

令 $\displaystyle \varepsilon\to 0^+$ 即得

$$\begin{aligned} \left|\begin{array}{cccccccccc}A&B\\\\ C&D\end{array}\right|=|AD-CB|. \tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\\\mbox{zhangzujin.cn}\end{array}}\end{aligned}$$

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