基本的な行列の問題

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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