张祖锦常用结论08偶数阶方阵行列式的迅速降阶法

zhangzujin

手机查看请在浏览器中打开, 到了支付页面请截图, 并用支付宝或微信扫描之, 稍等后获得金钱, 即可购买. 偶偶因为网络问题充值不成功, 请与微信 pdezhang 联系, 发送论坛昵称与付款时间即可处理, 稍安勿躁. 购买后刷新网页才能正常显示数学公式.

# 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\} [纸质资料](https://mp.weixin.qq.com/s/ycnPCSqWFlThEnq9ZZ6gBQ)/[答疑](https://mp.weixin.qq.com/s/JGYZG5rsshf7Z2Amo2di8A)/[pdf1](https://mp.weixin.qq.com/s/Pt6\_h5MqtomrUDYiPEwkxg)/[pdf2](https://mp.weixin.qq.com/s/dWvpeJFKnFr0WYPoidXXMA) / (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\} 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)/