张祖锦常用结论02摄动法示例
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设 $\displaystyle |A|=|a\_\{ij\}|$ 是一个 $\displaystyle n$ 阶行列式, $\displaystyle A\_\{ij\}$ 是它的 $\displaystyle (i,j)$ 元素的代数余子式, 求证:
\begin\{aligned\} \left|\begin\{array\}\{cccccccccc\}A&x\\\\ y^\mathrm\{T\}&1\end\{array\}\right|=|A|-\sum\_\{i,j=1\}^n A\_\{ij\}x\_iy\_j, \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\}
其中
\begin\{aligned\} x=(x\_1,\cdots,x\_n)^\mathrm\{T\}, y=(y\_1,\cdots,y\_n)^\mathrm\{T\}. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\}
[纸质资料](https://mp.weixin.qq.com/s/ycnPCSqWFlThEnq9ZZ6gBQ)/[答疑](https://mp.weixin.qq.com/s/JGYZG5rsshf7Z2Amo2di8A)/(https://mp.weixin.qq.com/s/Pt6\_h5MqtomrUDYiPEwkxg)/(https://mp.weixin.qq.com/s/dWvpeJFKnFr0WYPoidXXMA) /
(1)、 当 $\displaystyle A$ 可逆时,
\begin\{aligned\} \mbox\{左端\}&=\left|\begin\{array\}\{cccccccccc\}A&x\\\\ y^\mathrm\{T\}&1\end\{array\}\right|=|A|\left(1-y^\mathrm\{T\} A^\{-1\}x\right) =|A|-y^\mathrm\{T\} A^\star x =|A|-\sum\_\{i,j\}y\_j A\_\{ij\}x\_i. \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 (注意:任意数域都包含有理数域)
\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&x\\\\ y^\mathrm\{T\}&1\end\{array\}\right|=|A^\varepsilon|-\sum\_\{i,j\}A^\varepsilon\_\{ij\}x\_iy\_j. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\}
令 $\displaystyle \varepsilon\to 0^+$ 即知
\begin\{aligned\} \left|\begin\{array\}\{cccccccccc\}A&x\\\\ y^\mathrm\{T\}&1\end\{array\}\right|=|A|-\sum\_\{i,j=1\}^n A\_\{ij\}x\_iy\_j. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\}
跟锦数学微信公众号. [在线资料](https://mp.weixin.qq.com/s/F-TU-uzeo3EjxI5LzjUvRw)/[公众号](https://mp.weixin.qq.com/s/pdC49P5WZXTEpRBa0JBfow)/
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