2.4* 矩阵公式总结

操作顺序可交换

$$({\mathit{A}}^{?}{)}^{?}=({\mathit{A}}^{?}{)}^{?}⟹\{\begin{array}{l}({\mathit{A}}^{\ast }{)}^{-1}=({\mathit{A}}^{-1}{)}^{\ast }\\ ({\mathit{A}}^{\mathrm{T}}{)}^{\ast }=({\mathit{A}}^{\ast }{)}^{\mathrm{T}}\\ ({\mathit{A}}^{-1}{)}^{\mathrm{T}}=({\mathit{A}}^{\mathrm{T}}{)}^{-1}\end{array}$$

整体操作需对调

$$(\mathit{A}\mathit{B}{)}^{?}={\mathit{B}}^{?}{\mathit{A}}^{?}⟹\{\begin{array}{l}(\mathit{A}\mathit{B}{)}^{-1}={\mathit{B}}^{-1}{\mathit{A}}^{-1}\\ (\mathit{A}\mathit{B}{)}^{\mathrm{T}}={\mathit{B}}^{\mathrm{T}}{\mathit{A}}^{\mathrm{T}}\\ (\mathit{A}\mathit{B}{)}^{\ast }={\mathit{B}}^{\ast }{\mathit{A}}^{\ast }\end{array}$$

拓展：$(\mathit{A}\mathit{B}\mathit{C}{)}^{?}={\mathit{C}}^{?}{\mathit{B}}^{?}{\mathit{A}}^{?}$

重复操作会还原

$$({\mathit{A}}^{?}{)}^{?}=\mathit{A}⟹\{\begin{array}{l}({\mathit{A}}^{\mathrm{T}}{)}^{\mathrm{T}}=\mathit{A}\\ ({\mathit{A}}^{-1}{)}^{-1}=\mathit{A}\end{array}$$

WARNING

两次取伴随矩阵不会还原：$({\mathit{A}}^{\ast }{)}^{\ast }\ne \mathit{A}$

转置的优良性

$${\mathit{A}}^{\mathrm{T}}⟹\{\begin{array}{l}|{\mathit{A}}^{\mathrm{T}}|=|\mathit{A}|\\ (k\mathit{A}{)}^{\mathrm{T}}=k{\mathit{A}}^{\mathrm{T}}\\ (\mathit{A}+\mathit{B}{)}^{\mathrm{T}}={\mathit{A}}^{\mathrm{T}}+{\mathit{B}}^{\mathrm{T}}\end{array}$$

WARNING

$$\{\begin{array}{l}(\mathit{A}+\mathit{B}{)}^{-1}\ne {\mathit{A}}^{-1}+{\mathit{B}}^{-1}\\ (\mathit{A}+\mathit{B}{)}^{\ast }\ne {\mathit{A}}^{\ast }+{\mathit{B}}^{\ast }\end{array}$$

逆矩阵的定义

$$\mathit{A}{\mathit{A}}^{-1}=\mathit{E}⟹\{\begin{array}{rl}& |{\mathit{A}}^{-1}|=\frac{1}{|\mathit{A}|}\\ & (k\mathit{A}{)}^{-1}=\frac{1}{k}{\mathit{A}}^{-1}\end{array}$$

伴随矩阵的推导

$${\mathit{A}}^{\ast }=|\mathit{A}|{\mathit{A}}^{-1}⟹\{\begin{array}{l}(k\mathit{A}{)}^{\ast }=k^{n-1}{\mathit{A}}^{\ast }\\ |{\mathit{A}}^{\ast }|=|\mathit{A}{|}^{n-1}\\ ({\mathit{A}}^{\ast }{)}^{\ast }=|\mathit{A}{|}^{n-2}\mathit{A}\end{array}$$

${\mathit{A}}^{\ast }=|\mathit{A}|{\mathit{A}}^{-1}$ 这一点要牢牢记住。