# 基本的な行列の問題

## ■問題

$A=\left(\begin{array}{cccc}1& 2& 3& 4\\ 2& 3& 4& 6\\ 3& 4& 6& 9\\ 4& 6& 9& 10\end{array}\right)$

## ■答

${A}^{-1}=\left(\begin{array}{cccc}-\frac{7}{3}& 0& \frac{2}{3}& \frac{1}{3}\\ 0& 3& -2& 0\\ \frac{2}{3}& -2& \frac{2}{3}& \frac{1}{3}\\ \frac{1}{2}& 0& \frac{1}{3}& -\frac{1}{3}\end{array}\right)$

## ■計算

まずは$\left|A\right|$を求める．

$\left|A\right|$ $=\left|\begin{array}{cccc}1& 2& 3& 4\\ 2& 3& 4& 6\\ 3& 4& 6& 9\\ 4& 6& 9& 10\end{array}\right|$

$=\left|\begin{array}{cccc}1& 2& 3& 4\\ 0& -1& -2& -2\\ 0& -2& -3& -3\\ 0& -2& -3& -6\end{array}\right|$

$=\left|\begin{array}{ccc}-1& -2& -2\\ -2& -3& -3\\ -2& -3& -6\end{array}\right|$

$=-\left|\begin{array}{ccc}1& 2& 2\\ 2& 3& 3\\ 2& 3& 6\end{array}\right|$

$=-\left|\begin{array}{ccc}1& 2& 2\\ 0& -1& -1\\ 0& -1& 2\end{array}\right|$

$=-\left|\begin{array}{cc}-1& -1\\ -1& 2\end{array}\right|$

$=-\left(-2-1\right)$

$=3$$\ne 0$

${\stackrel{˜}{a}}_{11}$$={\left(-1\right)}^{1+1}\left|\begin{array}{ccc}3& 4& 6\\ 4& 6& 9\\ 6& 9& 10\end{array}\right|$$=-\left|\begin{array}{ccc}1& 2& 3\\ 4& 6& 9\\ 6& 9& 10\end{array}\right|$$=-\left|\begin{array}{ccc}1& 2& 3\\ 0& -2& -3\\ 0& -3& -8\end{array}\right|$$=-\left|\begin{array}{cc}2& 3\\ 3& 8\end{array}\right|$$=-7$

${\stackrel{˜}{a}}_{12}$$={\left(-1\right)}^{1+2}\left|\begin{array}{ccc}2& 4& 6\\ 3& 6& 9\\ 4& 9& 10\end{array}\right|$$=\left|\begin{array}{ccc}1& 2& 3\\ 3& 6& 9\\ 4& 9& 10\end{array}\right|$$=\left|\begin{array}{ccc}1& 2& 3\\ 0& 0& 0\\ 0& 1& -2\end{array}\right|$$=\left|\begin{array}{cc}0& 0\\ 1& -2\end{array}\right|$$=0$

${\stackrel{˜}{a}}_{13}$$={\left(-1\right)}^{1+3}\left|\begin{array}{ccc}2& 3& 6\\ 3& 4& 9\\ 4& 6& 10\end{array}\right|$$=2|\begin{array}{ccc}2& 3& 6\\ 3& 4& 9\\ 2& 3& 5\end{array}|$$=-2\left|\begin{array}{ccc}1& 1& 3\\ 0& 1& 0\\ 0& 1& -1\end{array}\right|$$=-2\left|\begin{array}{cc}1& 0\\ 1& -1\end{array}\right|$$=2$

${\stackrel{˜}{a}}_{14}$$={\left(-1\right)}^{1+4}\left|\begin{array}{ccc}2& 3& 4\\ 3& 4& 6\\ 4& 6& 9\end{array}\right|$$=\left|\begin{array}{ccc}1& 1& 2\\ 3& 4& 6\\ 4& 6& 9\end{array}\right|$$=\left|\begin{array}{ccc}1& 1& 2\\ 0& 1& 0\\ 0& 2& 1\end{array}\right|$$=\left|\begin{array}{cc}1& 0\\ 2& 1\end{array}\right|$$=1$

${\stackrel{˜}{a}}_{21}$$={\left(-1\right)}^{2+1}\left|\begin{array}{ccc}2& 3& 4\\ 4& 6& 9\\ 6& 9& 10\end{array}\right|$$=-6\left|\begin{array}{ccc}1& 1& 4\\ 2& 2& 9\\ 3& 3& 10\end{array}\right|$$=-6\left|\begin{array}{ccc}1& 1& 4\\ 0& 0& 1\\ 0& 0& -2\end{array}\right|$$=-6\left|\begin{array}{cc}0& 1\\ 0& -2\end{array}\right|$$=0$

${\stackrel{˜}{a}}_{22}$$={\left(-1\right)}^{2+2}\left|\begin{array}{ccc}1& 3& 4\\ 3& 6& 9\\ 4& 9& 10\end{array}\right|$$=\left|\begin{array}{ccc}1& 3& 4\\ 0& -3& -3\\ 0& -3& -6\end{array}\right|$$=\left|\begin{array}{cc}-3& -3\\ -3& -6\end{array}\right|$$=9\left|\begin{array}{cc}1& 1\\ 1& 2\end{array}\right|$$=9$

${\stackrel{˜}{a}}_{23}$$={\left(-1\right)}^{2+3}\left|\begin{array}{ccc}1& 2& 4\\ 3& 4& 9\\ 4& 6& 10\end{array}\right|$$=-\left|\begin{array}{ccc}1& 2& 4\\ 0& -2& -3\\ 0& -2& -6\end{array}\right|$$=-\left|\begin{array}{cc}-2& -3\\ -2& -6\end{array}\right|$$=-6\left|\begin{array}{cc}1& 1\\ 1& 2\end{array}\right|$$=-6$

${\stackrel{˜}{a}}_{24}$$={\left(-1\right)}^{2+4}\left|\begin{array}{ccc}1& 2& 3\\ 3& 4& 6\\ 4& 6& 9\end{array}\right|$$=\left|\begin{array}{ccc}1& 2& 3\\ 0& -2& -3\\ 0& -2& -3\end{array}\right|$$=\left|\begin{array}{cc}-2& -3\\ -2& -3\end{array}\right|$$=6\left|\begin{array}{cc}1& 1\\ 1& 1\end{array}\right|$$=0$

${\stackrel{˜}{a}}_{31}$$={\left(-1\right)}^{3+1}\left|\begin{array}{ccc}2& 3& 4\\ 3& 4& 6\\ 6& 9& 10\end{array}\right|$$=-2\left|\begin{array}{ccc}1& 1& 1\\ 3& 4& 3\\ 6& 9& 5\end{array}\right|$$=-2\left|\begin{array}{ccc}1& 1& 1\\ 0& 1& 0\\ 0& 3& -1\end{array}\right|$$=-2\left|\begin{array}{cc}1& 0\\ 3& -1\end{array}\right|$$=-2$

${\stackrel{˜}{a}}_{32}$$={\left(-1\right)}^{3+2}\left|\begin{array}{ccc}1& 3& 4\\ 2& 4& 6\\ 4& 9& 10\end{array}\right|$$=-2\left|\begin{array}{ccc}1& 3& 4\\ 1& 2& 3\\ 4& 9& 10\end{array}\right|$$=-2\left|\begin{array}{ccc}1& 3& 4\\ 0& -1& -1\\ 0& -3& -6\end{array}\right|$$=-2\left|\begin{array}{cc}-1& -1\\ -3& -6\end{array}\right|$$=-6$

${\stackrel{˜}{a}}_{33}$$={\left(-1\right)}^{3+3}\left|\begin{array}{ccc}1& 2& 4\\ 2& 3& 6\\ 4& 6& 10\end{array}\right|$$=2\left|\begin{array}{ccc}1& 2& 4\\ 2& 3& 6\\ 2& 3& 5\end{array}\right|$$=2\left|\begin{array}{ccc}1& 2& 4\\ 0& -1& -2\\ 0& -1& -3\end{array}\right|$$=2\left|\begin{array}{cc}-1& -2\\ -1& -3\end{array}\right|$$=2$

${\stackrel{˜}{a}}_{34}$$={\left(-1\right)}^{3+4}\left|\begin{array}{ccc}1& 2& 3\\ 2& 3& 4\\ 4& 6& 9\end{array}\right|$$=-\left|\begin{array}{ccc}1& 2& 3\\ 0& -1& -2\\ 0& -2& -3\end{array}\right|$$=-|\begin{array}{cc}-1& -2\\ -2& -3\end{array}|$$=-\left|\begin{array}{cc}-1& -2\\ -2& -3\end{array}\right|$$=1$

${\stackrel{˜}{a}}_{41}$$={\left(-1\right)}^{4+1}\left|\begin{array}{ccc}2& 3& 4\\ 3& 4& 6\\ 4& 6& 9\end{array}\right|$$=\left|\begin{array}{ccc}1& 1& 2\\ 3& 4& 6\\ 4& 6& 9\end{array}\right|$$=\left|\begin{array}{ccc}1& 1& 2\\ 0& 1& 0\\ 0& 2& 1\end{array}\right|$$=\left|\begin{array}{cc}1& 0\\ 2& 1\end{array}\right|$$=1$

${\stackrel{˜}{a}}_{42}$$={\left(-1\right)}^{4+2}\left|\begin{array}{ccc}1& 3& 4\\ 2& 4& 6\\ 3& 6& 9\end{array}\right|$$=3\left|\begin{array}{ccc}1& 3& 4\\ 2& 4& 6\\ 1& 2& 3\end{array}\right|$$=3\left|\begin{array}{ccc}1& 3& 4\\ 0& -2& -2\\ 0& -1& -1\end{array}\right|$$=3\left|\begin{array}{cc}-2& -2\\ -1& -1\end{array}\right|$$=0$

${\stackrel{˜}{a}}_{43}$$={\left(-1\right)}^{4+3}\left|\begin{array}{ccc}1& 2& 4\\ 2& 3& 6\\ 3& 4& 9\end{array}\right|$$=-\left|\begin{array}{ccc}1& 2& 4\\ 2& 3& 6\\ 3& 4& 9\end{array}\right|$$=-\left|\begin{array}{ccc}1& 2& 4\\ 0& -1& -2\\ 0& -2& -3\end{array}\right|$$=-\left|\begin{array}{cc}-1& -2\\ -2& -3\end{array}\right|$$=1$

${\stackrel{˜}{a}}_{44}$$={\left(-1\right)}^{4+4}\left|\begin{array}{ccc}1& 2& 3\\ 2& 3& 4\\ 3& 4& 6\end{array}\right|$$=\left|\begin{array}{ccc}1& 2& 3\\ 2& 3& 4\\ 3& 4& 6\end{array}\right|$$=\left|\begin{array}{ccc}1& 2& 3\\ 0& -1& -2\\ 0& -2& -3\end{array}\right|$$=\left|\begin{array}{cc}-1& -2\\ -2& -3\end{array}\right|$$=-1$

${A}^{-1}$$=\frac{1}{3}\left(\begin{array}{cccc}-7& 0& 2& 1\\ 0& 9& -6& 0\\ 2& -6& 2& 1\\ 1& 0& 1& -1\end{array}\right)$ $=\left(\begin{array}{cccc}-\frac{7}{3}& 0& \frac{2}{3}& \frac{1}{3}\\ 0& 3& -2& 0\\ \frac{2}{3}& -2& \frac{2}{3}& \frac{1}{3}\\ \frac{1}{3}& 0& \frac{1}{3}& -\frac{1}{3}\end{array}\right)$

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