张祖锦常用结论01行列式的迅速降阶法

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行列式的迅速降阶法

设 $\displaystyle A$ 是 $\displaystyle m\times n$ 矩阵, $\displaystyle B$ 是 $\displaystyle n\times m$ 矩阵, $\displaystyle E_k$ 表示 $\displaystyle k$ 阶单位矩阵, 则

$$\begin{aligned} \lambda^n\cdot |\lambda E_m-AB|=\lambda^m\cdot |\lambda E_n-BA|,\left(\mbox{$\lambda$ 是复数}\right). \tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\\\mbox{zhangzujin.cn}\end{array}}\end{aligned}$$

取 $\displaystyle \lambda=1$, 得到如下计算行列式的常用形式:

$$\begin{aligned} |E_m+AB|=|E_n+BA|. \tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\\\mbox{zhangzujin.cn}\end{array}}\end{aligned}$$

而如果 $\displaystyle m=1$, 则立马得到 $\displaystyle n$ 阶行列式的值!

纸质资料/答疑/pdf1/pdf2 /

(1)、 当 $\displaystyle \lambda=0$ 时, 结论自明.

(2)、 当 $\displaystyle \lambda\neq 0$　时, 由

$$\begin{aligned} \left(\begin{array}{cccccccccc} E_m&-A\\\\ 0&E_n\end{array}\right)\left(\begin{array}{cccccccccc} \lambda E_m&A\\\\ B&E_n\end{array}\right)\left(\begin{array}{cccccccccc} E_m&0\\\\ -B&E_n\end{array}\right)&=\left(\begin{array}{cccccccccc} \lambda E_m-AB&0\\\\ 0&E_n\end{array}\right),\\\\ \left(\begin{array}{cccccccccc} E_m&0\\\\ -\frac{1}{\lambda}B&E_n\end{array}\right)\left(\begin{array}{cccccccccc} \lambda E_m&A\\\\ B&E_n\end{array}\right)\left(\begin{array}{cccccccccc} E_m&-\frac{1}{\lambda}A\\\\ 0&E_n\end{array}\right)&=\left(\begin{array}{cccccccccc} \lambda E_m&0\\\\ 0&E_n-\frac{1}{\lambda }BA\end{array}\right) \tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\\\mbox{zhangzujin.cn}\end{array}}\end{aligned}$$

知

$$\begin{aligned} \left|\begin{array}{cccccccccc} \lambda E_m-AB&0\\\\ 0&E_n\end{array}\right|=\left|\begin{array}{cccccccccc} \lambda E_m&A\\\\ B&E_n\end{array}\right|=\left|\begin{array}{cccccccccc} \lambda E_m&0\\\\0&E_n-\frac{1}{\lambda }BA\end{array}\right|, \tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\\\mbox{zhangzujin.cn}\end{array}}\end{aligned}$$

$$\begin{aligned} |\lambda E_m-AB| =\lambda ^m \left|E_n-\frac{1}{\lambda }BA\right|=\lambda ^{m-n}|\lambda E_n-BA|. \tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\\\mbox{zhangzujin.cn}\end{array}}\end{aligned}$$

跟锦数学微信公众号. 在线资料/公众号/