逆行列

逆行列

逆行列について

$$\begin{array}{r}{\mathit{A}}^{-1}\mathit{A}=\mathit{A}{\mathit{A}}^{-1}=\mathit{I}\end{array}$$ となる行列${\mathit{A}}^{-1}$を逆行列という．

余因子行列と元の行列の積

$$\begin{array}{r}\tilde{\mathit{A}}\mathit{A}=\mathit{A}\tilde{\mathit{A}}=\left|\mathit{A}\right|\mathit{I}\dots \tilde{\mathit{A}}:\mathrm{行}\mathrm{列}\mathit{A}\mathrm{の}\mathrm{余}\mathrm{因}\mathrm{子}\mathrm{行}\mathrm{列}\end{array}$$

余因子行列を元に逆行列を求める

$$\begin{array}{rclrc}\tilde{\mathit{A}}\mathit{A}& =& \mathit{A}\tilde{\mathit{A}}& =& \left|\mathit{A}\right|\mathit{I}\\ \frac{\tilde{\mathit{A}}}{\left|\mathit{A}\right|}\mathit{A}& =& \mathit{A}\frac{\tilde{\mathit{A}}}{\left|\mathit{A}\right|}& =& \mathit{I}\\ {\mathit{A}}^{-1}& =& \frac{\tilde{\mathit{A}}}{\left|\mathit{A}\right|}\end{array}$$

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例 2x2行列の逆行列

$$\begin{array}{r}\\ {\mathit{A}}_{2\times 2}& =& \left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\\ {\tilde{\mathit{A}}}_{2\times 2}& =& \left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]\\ {\mathit{A}}_{2\times 2}^{-1}& =& \frac{1}{\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]\\ & =& \frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]\end{array}$$