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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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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