1739 2011 21-36 21 ( (Majima, Hideyuki(Ochanomizu University 1 1960 2Jl 28 50 2009 2 28 2010 2 28 50 ( WEBPAGE http://mathsocjp/meeting/takagi50/ 50 2009 2010 20 $\rangle$ (1898 (1904 2 (1949 (1970 $ $ 1 $\rangle$ (1935 2 (1970 (1933 ; : (1938 $(1943$ (1961 2009 8 2010 2 50 2 (1961( 1969 5 30 10 40 1 ( 3 (1961 11 $p$ 1933 : 1938 $(1943$ $(12p$ $(1961$ $$ $(11$ $p$ [1955
22 1986 (1961 2010 9 15 20 ( ( \S 40 \S 53 [3] 3 ( 1898 Theory ( of determinants $($ - $1919$ $($ $1930$ $($ $1948$ ( 1965 ( Weber Determinant [ ( 1678 : (Leibnitz (1678 (Cramer (1750 (\S 9 $(Cauchy$ $1789-1857$ $(Jacobi$ 1804 1851 $$ $(Cayley$ 1821 1895 ( 1678 :
23 Leibnitz (1678 Cramer (1750 Cauchy $(1815$ Jacobi (1841 1 [ (Leibnitz (1678 (Leibnitz de 1 Hosptal 1693 ( (Leibnitz 1678 $\angle $ ( $3$ $\cdots$ $\cdot\cdot\cdot$ ( (Leibnitz (1678 2 200 (1907 1908 1910 20 3 ( 5 4 ( 2 2 2 3 3 41 3 3 2 2 3 3 $\hslash\backslash$ 3, 3 3 $\{\begin{array}{l}a_{1}x+b_{1}y+c_{1}z=k_{1} (1a_{2}x+b_{2}y+c_{2}z=k_{2} (2a_{3}x+b_{3}y+c_{3}Jz=k_{3}\ldots(3\end{array}$ (2 (3 $y$ $z$
$ _{a_{3}}^{2\cross}a_{2}b_{3}+(3\cross(-b_{2}b_{2 x- _{b_{3}c_{3}}^{b_{2}c_{2}}}b_{3} z= k_{3}k_{2}$ 24 $b_{3}b_{2} \ldots(4$ $(2 \cross(;_{3_{c_{\backslash }}^{+}}a_{2}c_{2}a_{33}_{x+}^{3\cross} _{b_{3}}^{-c_{2}}b_{2}c_{3}c_{2} y= k_{3}k_{2}$ $r_{\vee 3}c_{2} \ldots(5$ (1 $y$ $z$ (1 $\cross \begin{array}{ll}b_{2} c_{2}b_{3} c_{3}\end{array} +(5\cross(-b_{1}+(4\cross c_{1}$ $\{a_{1} \begin{array}{ll}b_{2} c_{2}b_{3} c_{3}\end{array} -b_{1} \begin{array}{ll}a_{2} c_{2}a_{3} c_{3}\end{array} +c_{1} \begin{array}{ll}a_{2} b_{2}a_{3} b_{3}\end{array} \}x$ $=\{k_{1} \begin{array}{ll}b_{2} c_{2}b_{3} c_{3}\end{array} -b_{1} \begin{array}{ll}k_{2} c_{2}k_{3} c_{3}\end{array} +c_{1} \begin{array}{ll}k_{2} b_{2}k_{3} b_{3}\end{array} \}$ (1 $\cross \begin{array}{ll}b_{2} c_{2}b_{3} c_{3}\end{array} +(2\cross(- \begin{array}{ll}b_{l} c_{1}b_{3} c_{3}\end{array} +(3\cross \begin{array}{ll}b_{l} c_{1}b_{2} c_{2}\end{array} $, $\{a_{1} \begin{array}{ll}b_{2} c_{2}b_{3} c_{3}\end{array} -a_{2} \begin{array}{ll}b_{1} c_{1}b_{3} c_{3}\end{array} +a_{3} \begin{array}{ll}b_{1} c_{1}b_{2} c_{2}\end{array} \}x$ $+\{b_{1} \begin{array}{ll}b_{2} c_{2}b_{3} c_{3}\end{array} -b_{2} \begin{array}{ll}b_{1} c_{1}b_{3} c_{3}\end{array} +b_{3} \begin{array}{ll}b_{l} c_{1}b_{2} c_{2}\end{array} \}y$ $+\{c_{1} \begin{array}{ll}b_{2} c_{2}b_{3} c_{3}\end{array} -c_{2} \begin{array}{ll}b_{1} c_{1}b_{3} c_{3}\end{array} +c_{3} \begin{array}{ll}b_{1} c_{l}b_{2} c_{2}\end{array} \}z$ $=\{k_{1} \begin{array}{ll}b_{2} c_{2}b_{3} c_{3}\end{array} -k_{2} \begin{array}{ll}b_{1} c_{1}b_{3} c_{3}\end{array} +k_{3} \begin{array}{ll}b_{1} c_{1}b_{2} c_{2}\end{array} \}$ $a_{1} \begin{array}{ll}b_{2} c_{2}b_{3} c_{3}\end{array} -b_{1} \begin{array}{ll}a_{2} c_{2}a_{3} c_{3}\end{array} +c_{1} \begin{array}{ll}a_{2} b_{2}a_{3} b_{3}\end{array} $ $=$ $a_{1} \begin{array}{ll}b_{2} c_{2}b_{3} c_{3}\end{array} -a_{2} \begin{array}{ll}b_{1} c_{1}b_{3} c_{3}\end{array} +a_{3} \begin{array}{ll}b_{1} c_{1}b_{2} c_{2}\end{array} $ $b_{1} \begin{array}{ll}b_{2} c_{2}b_{3} c_{3}\end{array} -b_{2} \begin{array}{ll}b_{1} c_{l}b_{3} c_{3}\end{array} +b_{3} \begin{array}{ll}b_{1} c_{1}b_{2} c_{2}\end{array} $ $=$ $0$ $c_{1} \begin{array}{ll}b_{2} c_{2}b_{3} c_{3}\end{array} -c_{2} \begin{array}{ll}b_{1} c_{l}b_{3} c_{3}\end{array} +c_{3} \begin{array}{ll}b_{1} c_{1}b_{2} c_{2}\end{array} $ $=$ $0$ (1(2 (3 (2(1(3 1 2 (3(2(1, (1(3(2 1 3 2 3 $a_{1} \begin{array}{ll}b_{2} c_{2}b_{3} c_{3}\end{array} -a_{2} \begin{array}{ll}b_{1} c_{1}b_{3} c_{3}\end{array} +a_{3} \begin{array}{ll}b_{1} c_{1}b_{2} c_{2}\end{array} $ $=$ $-a_{2} \begin{array}{ll}b_{1} c_{1}b_{3} c_{3}\end{array} +b_{2} \begin{array}{ll}a_{1} c_{1}a_{3} c_{3}\end{array} -c_{2} \begin{array}{ll}a_{1} b_{1}a_{3} b_{3}\end{array} $
$=$ $a_{3} b_{2}b_{1}$ $c_{2}c_{1} -b_{3} a_{2}a_{1}$ $c_{2}c_{1} +c_{3} a_{2}a_{1}$ $b_{2}b_{1} $ $b_{3}b_{2}b_{1},$ 25 $y$ $a$ $b$ $-b_{1} \begin{array}{ll}a_{2} c_{2}a_{3} c_{3}\end{array} +b_{2} \begin{array}{ll}a_{1} c_{1}a_{3} c_{3}\end{array} -b_{3} \begin{array}{ll}a_{1} c_{1}a_{2} c_{2}\end{array} $ $=$ $a_{1} \begin{array}{ll}b_{2} c_{2}b_{3} c_{3}\end{array} -b_{1} \begin{array}{ll}a_{2} c_{2}a_{3} c_{3}\end{array} +c_{1} \begin{array}{ll}a_{2} b_{2}a_{i} b_{\theta}\end{array} $ $c_{1} \begin{array}{ll}a_{2} c_{2}a_{3} c_{3}\end{array} -c_{2}^{\backslash } \begin{array}{ll}a_{j} c_{1}a_{3} c_{3}\end{array} +c_{3} \begin{array}{ll}a_{1} c_{t}a_{2} c_{2}\end{array} =0$ $z$ $a$ $c$ $a_{1} \begin{array}{ll}b_{2} c_{2}b_{3} c_{3}\end{array} -b_{1} \begin{array}{ll}a_{2} t_{2}a_{3} c_{3}\end{array} +c_{1} \begin{array}{ll}a_{2} b_{2}a_{3} b_{3}\end{array} $ $=$ $c_{1} \begin{array}{ll}a_{2} b_{2}a_{3} b_{\theta}\end{array} -c_{2} \begin{array}{ll}a_{1} b_{1}a_{3} b_{3}\end{array} +c_{3} \begin{array}{ll}a_{1} b_{1}a_{2} b_{2}\end{array} $ $a_{1} \begin{array}{ll}b_{2} c_{2}b_{3} c_{i3}\end{array} -a_{2} \begin{array}{ll}b_{1} c_{1}b_{3} c_{3}\end{array} +as \begin{array}{ll}b_{1} c_{1}b_{2} c_{2}\end{array} $ ( 1 $=$ $-b_{1} \begin{array}{ll}a_{2} c_{2}\alpha_{3} c_{8}\end{array} +b_{2} \begin{array}{ll}a_{\ddagger} c_{1}a_{3} c_{3}\end{array} -b_{3} \begin{array}{ll}a_{1} c_{1}a_{2} c_{2}\end{array} $ ( 2 $=$ $c_{1} \begin{array}{ll}a_{2} b_{2}a_{3} b_{3}\end{array} -c_{2} \begin{array}{ll}a_{1} b_{1}a_{3} b_{3}\end{array} +c_{3} \begin{array}{ll}a_{1} b_{1}a_{2} b_{2}\end{array} $ ( 3 } $=$ $a_{1} \begin{array}{ll}b_{2} c_{2}b_{i} c_{3}\end{array} -b_{1} \begin{array}{ll}a_{2} c_{2}a_{3} c_{3}\end{array} +c_{1} \begin{array}{ll}a_{2} b_{2}a_{3} b_{3}\end{array} $ ( 1 $=$ $-a_{2} \begin{array}{ll}b_{1} c_{1}b_{3} c_{3}\end{array} +b_{2} \begin{array}{ll}a_{l} c_{1}a_{3} c_{3}\end{array} -c_{2} \begin{array}{ll}a_{1} b_{1}a_{3} b_{3}\end{array} $ ( 2 $=$ $a_{3} \begin{array}{ll}b_{1} c_{1}b_{2} c_{2}\end{array} -b_{3} \begin{array}{ll}a_{1} c_{1}a_{2} C\mathfrak{g}\end{array} +c_{3} \begin{array}{ll}a_{1} b_{1}a2 b_{2}\end{array} $ ( 3 $a_{1}b_{2}c_{3}+a_{2}b_{3}c_{1}+a_{\theta}b_{1}c_{2}-a_{1}bs\epsilon_{2}-a_{2}b_{1}c_{3}-a_{3}b_{2}c_{1}$ 3 $[o_{3}tt_{2}o_{1}$ $c_{3}c_{2}c_{1}]$,
$x= \frac{ _{k_{3},b_{3},c_{3}}^{k_{1},b_{1}c_{1}}k_{2},b_{2},c_{2} }{ _{a_{3} b_{3},c_{3}}^{a,b,c}a_{2}1b_{2}1c_{2}1 }$, $b_{3}b_{2}b_{1}$ b_{1} c_{1}a_{2\backslash } b_{2} c_{2}(l_{3} b_{3} c_{3}\end{array}\}$ \frac{ \begin{array}{lll}a_{1} k_{1} c_{1}a_{2} k_{2} c_{2}a_{3} k_{3} c_{3}\end{array} }{1_{a_{3},b_{3},c_{3}}^{a,b,c}a_{2},b_{2},c_{21}111}$ $z= \frac{ _{a_{3},b_{3},k_{3}}^{a_{1},b_{1},k_{1}}a_{2},b_{2},k_{2} }{ \begin{array}{lll}a_{1} b_{1} c_{l}a_{2} b_{2} c_{2}a_{3} b_{3} c_{3}\end{array} }$ 26 $\det\{\begin{array}{lll}a_{1} $ \begin{array}{lll}a_{1} b_{1} c_{1}a_{2} b_{2} c_{2}a_{3} b_{3} c_{3}\end{array} $, $ \begin{array}{lll}b_{1} b_{l} c_{1}b_{2} b_{2} c_{\prime 2}b_{3} b_{3} c_{3}\end{array} = \begin{array}{lll}c_{1} b_{l} c_{1}c_{2} b_{2} c_{2}c_{3} b_{3} c_{3}\end{array} = \begin{array}{lll}a_{1} c_{1} c_{l}a_{2} c_{2} c_{2}a_{3} c_{3} c_{3}\end{array} =0$ 3 2 O $ a_{3 }a_{2}a_{1}$ $c_{3}c_{1}c_{2} \neq 0$,, (1(2(3 Cramer $y= 42 3 $(\begin{array}{lll}a_{1} b_{1} c_{1}a_{2} b_{2} c_{2}a_{3} b_{3} c_{3}\end{array}$ 3 $ \begin{array}{lll}a_{1} b_{1} c_{1}a_{2} b_{2} c_{2}a_{3} b_{3} c_{3}\end{array} $ $a_{1}b_{2}c_{3}+a_{2}b_{3}c_{1}+a_{3}b_{1}c_{2}-a_{1}b_{3}c_{2}-a_{2}b_{1}c_{3}-a_{3}b_{2}c_{1}$ $\circ a_{1} \begin{array}{ll}b_{2} c_{2}b_{3} c_{3}\end{array} -a_{2} \begin{array}{ll}b_{1} c_{1}b_{3} c_{3}\end{array} +a_{3} \begin{array}{ll}b_{1} c_{1}b_{2} c_{2}\end{array} $ ( $17^{1J}$ $a_{1} \begin{array}{ll}b_{2} $-b_{1} \begin{array}{ll}a_{2} $-a_{2} \begin{array}{ll}b_{1} c_{2}b_{3} c_{3}\end{array} -b_{1} \begin{array}{ll}a_{2} c_{2}a_{3} c_{3}\end{array} +c_{1} \begin{array}{ll}a_{2} b_{2}a_{3} b_{3}\end{array} $ ( 1 c_{2}a_{3} c_{3}\end{array} +b_{2} \begin{array}{ll}a_{1} c_{1}a_{3} c_{3}\end{array} -b_{3} \begin{array}{ll}a_{1} c_{1}a_{2} c_{2}\end{array} $ ( 2 c_{1}b_{3} c_{3}\end{array} +b_{2} \begin{array}{ll}a_{1} c_{1}a_{3} c_{3}\end{array} -c_{2} \begin{array}{ll}a_{1} b_{1}a_{3} b_{3}\end{array} $ ( 2
$(F^{1}\rfloor 2 \begin{array}{lll}\lambda a_{1} b_{1} c_{1}\lambda a_{2} b_{2} c_{2}\lambda a_{3} b_{3} c_{3}\end{array} = \begin{array}{lll}a_{1} \lambda b_{1} c_{1}a_{2} \lambda b_{2} c_{2}a_{3} \lambda b_{3} c_{3}\end{array} = \begin{array}{lll}a_{1} b_{1} \lambda c_{1}a_{2} b_{2} \lambda c_{2}a_{3} b_{3} \lambda c_{3}\end{array} =\lambda \begin{array}{lll}a_{1} b_{1} c_{1}a_{2} b_{2} c_{2}a_{3} b_{3} c_{3}\end{array} $ a_{1} \lambda b_{1} \lambda c_{1}a_{2} b_{2} c_{2}a_{3} b_{3} c_{3}\end{array} = \begin{array}{lll}a_{1} b_{1} c_{1}\lambda a_{2} \lambda b_{2} \lambda c_{2}a_{3} b_{3} c_{3}\end{array} $ $= \begin{array}{lll}a_{1} b_{1} c_{1}a_{2} b_{2} c_{2}\lambda a_{3} \lambda b_{3} \lambda c_{3}\end{array} =\lambda \begin{array}{lll}a_{1} b_{1} c_{1}a_{2} b_{2} c_{2}a_{3} b_{3} c_{3}\end{array} $ 27 $c_{1} a_{3}a_{2}$ $a_{3} \begin{array}{ll}b_{1} $b_{3}b_{2} -c_{2} a_{3}a_{1}$ $b_{3}b_{1} +c_{3} a_{2}a_{1}$ $b_{2}b_{1} $ ( 3 c_{l}b_{2} r_{\prime 2}\end{array} -b_{3} \begin{array}{ll}a_{1} c_{1}a_{2} c_{2}\end{array} +c_{3} \begin{array}{ll}a_{1} b_{l}a_{2} b_{2}\end{array} $ ( 3 2 ( 1 $ \begin{array}{lll}a_{1} b_{1} c_{1}a_{2} b_{2} c_{2}a_{3} b_{3} c_{3}\end{array} =- \begin{array}{lll}b_{1} a] c_{1}b_{2} a_{2} c_{2}b_{3} a_{3} c_{3}\end{array} =- \begin{array}{lll}c_{1} b_{1} a_{1}c_{2} b_{2} a_{2}c_{3} b_{3} a_{3}\end{array} =- \begin{array}{lll}a_{1} c_{1} b_{1}a_{2} c_{2} b_{2}a_{3} c_{3} b_{3}\end{array} $ ( 1 $ \begin{array}{lll}a_{1} b_{1} c_{1}a_{2} b_{2} c_{2}a_{3} b_{3} c_{3}\end{array} =- \begin{array}{lll}a_{2} b_{2} c_{2}a_{1} b_{1} c_{1}a_{3} b_{3} c_{3}\end{array} =- \begin{array}{lll}a_{3} b_{3} c_{3}a_{2} b_{2} c_{2}a] b_{1} c_{1}\end{array} =- \begin{array}{lll}a_{1} b_{1} c_{1}a_{3} b_{3} c_{3}a_{2} b_{2} c_{2}\end{array} $ 2 0 ( 2 $ \begin{array}{lll}\lambda ( 3 $\{\begin{array}{l}[matrix]=[matrix]+[matrix][matrix]=[matrix]+[matrix][matrix]=[matrix]+[matrix]\end{array}$ ( 3 $\{\begin{array}{l}[matrix]=[matrix]+[matrix][matrix]=[matrix]+[matrix][matrix]=[matrix]+[matrix]\end{array}$
$b_{3}b_{2}b_{1}$ $c_{3}c_{2}c1$ $b_{3}b_{2}b_{1}$ $c_{3}c_{2}c1$ 28 (0 $ \begin{array}{lll}l 0 00 l 00 0 l\end{array} =1$ 3 ( $M(3, R$ $R$ ( $F$ (0 ( 1 ( 2 ( 3 (0 ( 1 ( 2 ( 3 ( 1( 2( 3 ( 1 ( 2 ( 3 3 ( $M(3, R$ $R$ ( $F(\begin{array}{lll}a_{1} b_{1} c_{1}a_{2} b_{2} c_{2}a_{3} b_{3} c_{3}\end{array}=f(\begin{array}{lll}l 0 00 l 00 0 1\end{array} a_{3}a_{1}a_{2}$ (0 $F(\begin{array}{lll}a_{1} b_{1} c_{1}a_{2} b_{2} c_{2}a_{3} b_{3} c_{3}\end{array}=f(\begin{array}{lll}l 0 00 1 00 0 l\end{array} \begin{array}{lll}a_{1} b_{l} c_{1}a_{2} b_{2} c_{2}a_{3} b_{3} c_{3}\end{array} = a_{2}a_{3}a_{1}$ 5 ( ( ( 1683 $+$ (? ( 1 2 2 3 3 4 ( ( 3 3 ( 3 2 $x_{11}+x_{12}y=0,$ $x_{21}+x_{22}y=0$ $y$ 1 1 $\det_{2}\{\begin{array}{ll}x_{11} x_{12}x_{21} x_{22}\end{array}\}= x_{11}x_{21}$ $x_{22}x_{12}$ $=$ $+x_{11}x_{22}$ $-x_{21}x_{12}$ 0
$+(5\cross(-x_{12}+(4\cross 29 $y$ 2 1 $\{\begin{array}{l}x_{11}+x_{12}y=0 (1x_{21}+x_{22}y=0 (2\end{array}$ (1 (2 $y$ (1 $\cross x_{22}+(2\cross(-x_{12}$ $(+x_{11}x_{22}-x_{21}x_{12}+(+x_{12}x_{22}-x_{22}x_{12}y$ $=$ $0$ 1 $y$ $[x_{22}-x_{12}]\{\begin{array}{ll}x_{11} x_{12}x_{21} x_{22}\end{array}\}=[+x_{11}x_{22}-x_{2\mathfrak{l}}x_{12}$ $+x_{12}x_{22}-x_{22}x_{12}]$ 3 $y$ 3 2 $\{\begin{array}{l}x_{11}+x_{12}y+x_{13}y^{2}=0 (1x_{21}+x_{22}y+x_{23}y^{2}=0 (2x_{31}+x_{32}y+x_{33}y^{2}=0 (3\end{array}$ (2 (3 $y$ (2 $\cross x_{32}+(3\cross(-x_{22}$ $x_{31}x_{21}$ $x_{32}x_{22} - \begin{array}{ll}x_{22} x_{23}x_{32} x_{33}\end{array} y^{2}=0\ldots(4$ $x_{32}x_{22}$ $x_{31}x_{21}$ $ + \begin{array}{ll}x_{22} x_{23}x_{32} x_{33}\end{array} y^{2}=0\ldots(4$ (2 (3 (2 $y^{2}$ $\cross x_{33}+(3\cross(-x_{23}$ $x_{31}x_{21}$ $x_{33}x_{23} + \begin{array}{ll}x_{22} x_{23}x_{32} x_{33}\end{array} y=0\ldots(5$ (1 $\cross$ (1 $x_{22}$ $x_{32}$ $x_{23}$ $x_{33}$ x_{13}$ $y$ $y^{2}$ $x_{11}$ $x_{32}x_{22}$ $x_{33}x_{23} -x_{12} \begin{array}{ll}x_{21} x_{23}x_{31} x_{33}\end{array} +x_{13} x_{31}x_{21}$ $x_{32}x_{22} =0$ ( 2 $(+x_{11}x_{22}x_{33}-x_{11^{x}32^{x}23}+(+x_{31}x_{12}x_{23}-x_{21^{x}12^{x}33}+(+xxx-x_{31^{x}22^{x}13}=0$ (1, (2, (3 (1 $\cross \begin{array}{ll}x_{22} x_{23}x_{32} x_{33}\end{array} +[(2\cross x_{33}+(3\cross(-x_{23}]\cross(-x_{12}+[(2\cross x_{32}+(3\cross(-x_{22}]\cross x_{13}$, (1 $\cross \begin{array}{ll}x_{22} x_{23}x_{32} x_{33}\end{array} +(2\cross[x_{33}(-x_{12}+x_{32}x_{13}]+(3\cross[(-x_{23}(-x_{12}]+(-x_{22}x_{13}]$ 2 $(+x_{11}x_{22}x_{33}-x_{11^{x}32^{x}23}+(+x_{21}x_{32}x_{13}-x_{21^{x}12^{x}33}+(+x_{31}x_{12}x_{23}-x_{31^{x}22^{x}13}=0$
30 2 ( $x_{11} \begin{array}{ll}x_{22} x_{23}x_{32} x_{33}\end{array} +x_{21} \begin{array}{ll}x_{32} x_{33}x_{12} $x_{13}x_{33} +x_{31} x_{22}x_{12}$ x_{13}\end{array} +x_{31} x_{22}x_{12}$ $=0$ $x_{23}x_{13}$ $x_{11} \begin{array}{ll}x_{22} x_{23}x_{32} x_{33}\end{array} +x_{21}(- \begin{array}{ll}x_{12} x_{13}x_{32} x_{33}\end{array} +x_{31} x_{12}x_{22}$ $x_{23}x_{13}$ $=0$ 3 3 $\det$ 3 $\{\begin{array}{lll}x_{11} x_{l2} x_{13}x_{21} x_{22} x_{23}x_{31} x_{32} x_{33}\end{array}\}= x_{31}x_{21}xl1$ $x_{22}x_{32}x12$ $x_{23}x_{33}x13$ $=$ $+x_{11}x_{22}x_{33}+x_{21}x_{32}x_{13}+x_{31}x_{12}x_{23}$ $-x_{11}x_{32}x_{23}-x_{21}x_{12}x_{33}-x_{31}x_{22}x_{13}$ $-$ O $y^{2}$ 1, $y,$ $+\{x_{11}\cross[x_{22}x_{33}-x_{32}x_{23}\rfloor+x_{21}\cross[(-x_{12}x_{33}+x_{32}x_{13}]+x_{31}\cross[(-x_{12}(-x_{23}+(-x_{22}x_{13}]\}$ $+\{x_{12}\cross[x_{22}x_{33}-x_{32}x_{23}]+x_{22}\cross[(-x_{12}x_{33}+x_{32}x_{13}]+x_{32}\cross[(-x_{12}(-x_{23}+(-x_{22}x_{13}]\}y$ $+\{x_{13}\cross[x_{22}x_{33}-x_{32}x_{23}]+x_{23}\cross[(-x_{12}x_{33}+x_{32}x_{13} +x_{33}\cross[(-x_{12}(-x_{23}+(-x_{22}x_{13}]\}y^{2}$ $=$ $0$ 3 6 $u1=x_{11^{x}22^{x}33}$ $u2=x_{21^{x}32^{x}13}$, $?\iota 3=x_{31^{X}12^{X}23}$ $d1=x_{11}x_{32}x_{23}$, $d2=x_{21}x_{12}x_{33}$, $d3=x_{31}x_{22}x_{13}$ 6 $+u1-d1-d2+u3+u2-d3$, $+u1-d1+u2-d2+u3-d3$, $+u1+u2+u3-d1-d2-d3$ 3 $+u1-d3+u3-d1+u2-d2$, $+u1-d3-(+d1-u3+u2-d2$, 4 ( 4 det4 $\{\begin{array}{llll}x_{11} x_{l2} x_{13} x_{14}x_{21} x_{22} x_{23} x_{24}x_{31} x_{32} x_{33} x_{34}x_{41} x_{42} x_{43} x_{44}\end{array}\}=u1+u2+u3+d1+d2+d3$ $U1,$ $U2,$ $U3$ $D1$ $D2,$ $D3$ 4 $U1$ $=$ $+x_{11}x_{22}x_{33}x_{44}-x_{21}x_{32}x_{43}x_{14}+x_{31^{x}42^{x}13^{x}24}-x_{41^{x}12^{x}23^{x}34}$
$\{\begin{array}{l}x_{11}+x_{12}y+\cdot x_{13}y^{2}+x_{14}y^{3}=0\ldots(1x_{21}+x_{22}y+x_{23}y^{2}+x_{24}y^{3}=0\ldots(2x_{31}+x_{32}y+x_{33}y^{2}+x_{34}y^{3}=0\ldots(3x_{41}+x_{42}y+x_{43}y^{2}+x_{44}y^{3}=0\ldots(4\end{array}$ $x_{41}x_{21}x_{31}$ $x_{42}x_{32}x22$ $x_{43}x_{33}x23$ $x_{41}x_{31}x21$ $x_{43}x_{33}x23$ $x_{43}x_{33}x23$ $x_{44}x_{34}x24$ $x_{44}x_{34}x_{24}$ 31 $U2$ $=$ $-x_{11}x_{42}x_{33}x_{24}+x_{21}x_{12}x_{43}x_{34}-x_{31}x_{22}x_{13}x_{44}+x_{41}x_{32}x_{23}x_{14}$ $U3$ $=$ $-x_{11}x_{22}x_{43}x_{34}+x_{21}x_{32}x_{13}x_{44}-x_{31}x_{42}x_{23}x_{14}+x_{41}x_{12}x_{33}x_{24}$ $D1$ $=$ $+xxxx-x_{21}x_{12}x_{33}x_{44}+x_{31}x_{22}x_{43}x_{14}-x_{41^{x}32^{x}13^{x}24}$ $D2$ $=$ $+x_{11}x_{32}x_{43}x_{24}-x_{2j}x_{42}x_{13}x_{34}+x_{31}x_{12}x_{23}x_{44}-x_{41}x_{22}x_{33}x_{14}$ $D3$ $=$ $-x_{11}x_{32}x_{23}x_{44}+x_{21}x_{42}x_{33}x_{14}-x_{31}x_{12}x_{43}x_{24}+x_{41}x_{22}x_{13}x_{34}$ ( $SD(1234+SD(1342+SD(1423$, $SD(1234=U1+U2$, $SD(1342=D1+U3$, $SD(1423=D2+D3$ ( 4 6 4 $y$ 4 3 4 (2,(3,(4 3 $y,$ $y^{2},$ $y^{3}$ $y$ $0$ 1 2 3 3 ( 0 1 2 3 3 $x_{44}x_{34}x24 + \begin{array}{lll}x_{22} x_{23} x_{24}x_{32} x_{33} x_{34}x_{42} x_{43} x_{44}\end{array} y$ $=$ $0$ $x_{44}x_{34}x24 + x_{42}x_{32}x22$ $y^{2}$ $=$ $0$ $ \begin{array}{lll}x_{22} x_{23} x_{21}x_{32} x_{33} x_{31}x_{42} x_{43} x_{41}\end{array} + x_{42}x_{32}x22$ $y^{3}$ $=$ $0$ $y,$, $y^{2},$ $y^{3}$ $y^{2},$ $y^{3},$ $y$ $y^{3},$ $y,$, 2, 3, 4 3, 4, 2 4, 2, 3 $y^{2}$ ( 2 3 $x_{12},$ $x_{13},$ $x_{14}$ (1 ( 3 $ \begin{array}{lll}x_{22} x_{23} x_{24}x_{32} x_{33} x_{34}x_{42} x_{43} x_{44}\end{array} =0$
32 1 $y,$ $y^{2},$ $y^{3}$ $X11 \begin{array}{lll}x_{22} x_{23} x_{24}x_{32} x_{33} x_{34}x_{42} x_{43} x_{44}\end{array} -X12 \begin{array}{lll}x_{21} x_{23} x_{24}x_{31} x_{33} x_{34}x_{41} x_{43} x_{44}\end{array} -X13 \begin{array}{lll}x_{22} x_{21} x_{24}x_{32} x_{31} x_{34}x_{42} x_{41} x_{44}\end{array} -x_{14} \begin{array}{lll}x_{22} x_{23} x_{2l}x_{32} x_{33} x_{31}x_{42} x_{43} x_{4l}\end{array} =0$ 3 $x_{11} \begin{array}{lll}x_{22} x_{23} x_{24}x_{32} x_{33} x_{34}x_{42} x_{43} x_{44}\end{array} $ $x_{12}(+x_{21} \begin{array}{ll}x_{33} x_{34}x_{43} x_{44}\end{array} -x_{31} \begin{array}{ll}x_{23} x_{24}x_{43} x_{44}\end{array} +x_{41} \begin{array}{ll}x_{23} x_{24}x_{33} x_{34}\end{array} $ $x_{13}(-x_{21} \begin{array}{ll}x_{32} x_{34}x_{42} x_{44}\end{array} +x_{31} \begin{array}{ll}x_{22} x_{24}x_{42} x_{44}\end{array} -x_{41} \begin{array}{ll}x_{22} x_{24}x_{32} x_{34}\end{array} $ $-x_{14}(+x_{21} \begin{array}{ll}x_{32} x_{33}x_{42} x_{43}\end{array} -x_{31} \begin{array}{ll}x_{22} x_{23}x_{42} x_{43}\end{array} +x_{41} \begin{array}{ll}x_{22} x_{23}x_{32} x_{33}\end{array} $ $=$ $0$ $x_{21},$ $x_{31},$ $x_{41}$ g$\llcorner$ $x_{11} \begin{array}{lll}x_{22} x_{23} x_{24}x_{32} x_{33} x_{34}x_{42} x_{43} x_{44}\end{array} $ $-x_{21}(+x_{12} \begin{array}{ll}x_{33} x_{34}x_{43} x_{44}\end{array} -x_{13} \begin{array}{ll}x_{32} x_{34}x_{42} x_{44}\end{array} +x_{14} \begin{array}{ll}x_{32} x_{33}x_{42} x_{43}\end{array} $ $+x_{3}1(+x_{12} \begin{array}{ll}x_{23} x_{24}x_{43} x_{44}\end{array} -x_{13} \begin{array}{ll}x_{22} x_{24}x_{42} x_{44}\end{array} +x_{14} \begin{array}{ll}x_{22} x_{23}x_{42} x_{43}\end{array} $ $-x_{41}(+x_{12} \begin{array}{ll}x_{23} x_{24}x_{33} x_{34}\end{array} -x_{13} \begin{array}{ll}x_{22} x_{24}x_{32} x_{34}\end{array} +x_{14} \begin{array}{ll}x_{22} x_{23}x_{32} x_{33}\end{array} $ $=$ $0$ 3 4 $x_{11\cross} \begin{array}{lll}x_{22} x_{23} x_{24}x_{32} x_{33} x_{34}x_{42} x_{43} x_{44}\end{array} +x_{21}\cross(-1 \begin{array}{lll}x_{12} x_{13} x_{14}x_{32} x_{33} x_{34}x_{42} x_{43} x_{44}\end{array} +x_{31}\cross \begin{array}{lll}x_{12} x_{13} x_{14}x_{22} x_{23} x_{24}x_{42} x_{43} x_{44}\end{array} +x_{41}\cross(-1 \begin{array}{lll}x_{12} x_{13} x_{14}x_{22} x_{23} x_{24}x_{32} x_{33} x_{34}\end{array} =0$ $+x_{11}(+x_{22}x_{33}x_{44}-x_{22}x_{43}x_{34}-x_{32}x_{23}x_{44}+x_{32}x_{43}x_{24}+x_{42}x_{23}x_{34}-x_{42}x_{33}x_{24}$ $-x_{21}(+x_{12}x_{33}x_{44}-x_{12}x_{43}x_{34}-x_{13}x_{32}x_{44}+x_{13}x_{42}x_{34}+x_{14}x_{32}x_{43}-x_{14}x_{42}x_{33}$ $+x_{31}(+x_{12}x_{23}x_{44}-x_{12}x_{43}x_{24}-x_{13}x_{22}x_{44}+x_{13}x_{42}x_{24}+x_{14}x_{22}x_{43}-x_{14}x_{42}x_{23}$ $-x_{41}(+x_{12}x_{23}x_{34}-x_{12}x_{33}x_{24}-x_{13}x_{22}x_{34}+x_{13}x_{32}x_{24}+x_{14}x_{22}x_{33}-x_{14}x_{32}x_{23}$ $U1,$ $U3,$ $D3,$ $D2,$ $D1,$ $U2$ (2, (3, (4 (4
$x_{42}x_{32}x22$ $x_{43}x_{33}x23$ $x_{33}x_{23}x13$ $x_{33}x_{23}x13$ $x_{34}x_{24}x14$ $x_{34}x_{24}x14$ 33 3 $y,$ $y^{2},$ $y^{3}$ (2, (3, (4 $ \begin{array}{lll}x_{22} (1 x_{23} x_{24}x_{32} x_{33} x_{34}x_{42} x_{43} x_{44}\end{array} $ ( 3 1 $y,$ $y^{2},$ $y^{3}$ (1, (2, (3, (4 (1 $\cross \begin{array}{lll}x_{22} x_{23} x_{24}x_{32} x_{33} x_{34}x_{42} x_{43} x_{44}\end{array} +(2\cross(-1 \begin{array}{lll}x_{12} x_{13} x_{14}x_{32} x_{33} x_{34}x_{42} x_{43} x_{44}\end{array} +(3\cross \begin{array}{lll}x_{12} x_{13} x_{14}x_{22} x_{23} x_{24}x_{42} x_{43} x_{44}\end{array} +(4\cross(-1 x_{32}x_{22}x12$ 4 $x_{11\cross}$ $x_{44}x_{34}x24 +x_{21}\cross(-1 \begin{array}{lll}x_{1}\prime z x_{13} x_{14}x_{32} x_{33} x_{34}x_{42} x_{43} x_{44}\end{array} +x_{31\cross} \begin{array}{lll}x_{12} x_{13} x_{14}x_{22} x_{23} x_{24}x_{42} x_{43} x_{44}\end{array} +x_{41}\cross(-1 x_{32}x_{22}x12$ $X_{32}$ $x_{33}$ $X_{34} +x_{21}\cross(-1 x_{?2}$ $x_{33}$ $x_{?4} +x_{31}\cross x_{22}$ $x_{2?}$ $x_{24} +x_{41}\cross(-1 x_{22}$ $x_{23}$ $x_{24}$ $=0$ 1, 2, 3, 4, 2, 3, 4, 3, 4, 1, 4, 1, 2, 1, 2, 3, $x_{11}\cross \begin{array}{lll}x_{22} x_{23} x_{24}x_{32} x_{33} x_{34}x_{42} x_{43} x_{44}\end{array} +x_{21}\cross(-1 \begin{array}{lll}x_{32} x_{33} x_{34}x_{42} x_{43} x_{44}x_{12} x_{13} x_{14}\end{array} +x_{31}\cross \begin{array}{lll}x_{42} x_{43} x_{44}x_{12} x_{13} x_{14}x_{22} x_{23} x_{24}\end{array} +x_{41}\cross(-1 x_{32}x_{22}x12$ $x_{33}x_{23}x13$ $x_{24}x_{34}x14$ $=0$ 3 3 $UI,$ $D2,$ $DI;U3,$ $D3,$ $U2$ $+x_{11}(+x_{22}x_{33}x_{44}+x_{32}x_{43}x_{24}+x_{42^{x}23^{x}34}$ $+x_{11}(-x_{22}x_{43}x_{34}-x_{32}x_{23}x_{44}-x_{42}x_{33}x_{24}$ $-x_{21}(-x_{32}x_{13}x_{44}-x_{42}x_{33}x_{14}-x_{12^{x}43^{x}34}$ $-x_{21}(+x_{32}x_{43}x_{14}+x_{42}x_{13}x_{34}+x_{12}x_{33}x_{44}$ $+x_{31}(-x_{42}x_{23}x_{14}-x_{12}x_{43}x_{24}-x_{22}x_{13}x_{44}$ $+x_{31}(+x_{42}x_{13}x_{24}+x_{12}x_{23}x_{44}+x_{22}x_{43}x_{14}$ $-x_{41}(+x_{12}x_{23}x_{34}+x_{22}x_{33}x_{14^{\lrcorner}}-x_{32}x_{13}x_{24}$ $-x_{41}(-x_{12}x_{33}x_{24}-x_{22}x_{13}x_{34}-x_{32}x_{23}x_{14}$ 3 $U1,$ $D1,$ $D2;U3,$ $U2,$ $D3$ $+X_{11}(x_{422334}$ $+x_{11}(-x_{22}x_{43}x_{34}-x_{42}x_{33}x_{24}-x_{32^{x}23^{x}44}$ $-x_{21}(-x_{32}x_{13}x_{44}-x_{12}x_{43}x_{34}-x_{42^{x}33^{x}14}$ $-x_{21}(+x_{32}x_{43}x_{14}+x_{12}x_{33}x_{44}+x_{42^{x}13^{x}34}$ $+x_{31}(-x_{42}x_{23}x_{14}-x_{22}x_{13}x_{44}-x_{12^{x}43^{x}24}$ $+x_{31}(+xxx+x_{22}x_{43}x_{14}+x_{12^{x}23^{x}44}$
34 $-x_{41}(+x_{12}x_{23}x_{34}+x_{32}x_{13}x_{24}+x_{22}x_{33}x_{14}$ $-x_{41}(-x_{12}x_{33}x_{24}-x_{32}x_{23}x_{14}-x_{22}x_{13}x_{34}$ 4 $U1,$ $D1$, ( $U2,$ $D2,$ $U3,$ $D3$ $D2,$ $U3U2,$ $D3$ 2 2 $D2$ $U2$ $U1,$ $D1,$ $U2,$ $D2$, ; $U3,$ $D3$ 1 2 $+x_{12}(+x_{22}x_{33}x_{44}+x_{42}x_{23}x_{34}+x_{32}x_{43}x_{24}$ $+x_{12}(-x_{22}x_{43}x_{34}-x_{42}x_{33}x_{24}-x_{32}x_{23}x_{44}$ $-x_{22}(-x_{32}x_{13}x_{44}-x_{12}x_{43}x_{34}-x_{42}x_{33}x_{14}$ $-x_{22}(+x_{32}x_{43}x_{14}+x_{12}x_{33}x_{44}+x_{42}x_{13}x_{34}$ $+x_{32}(-x_{42}x_{23}x_{14}-x_{22}x_{13}x_{44}-x_{12}x_{43}x_{24}$ $+x_{32}(+x_{42}x_{13}x_{24}+x_{22}x_{43}x_{14}+x_{12}x_{23}x_{44}$ $-x_{42}(+x_{12}x_{23}x_{34}+x_{32}x_{13}x_{24}+x_{22}x_{33}x_{14}$ $-x_{42}(-x_{12}x_{33}x_{24}-x_{32}x_{23}x_{14}-x_{22}x_{13}x_{34}$ $x_{12}x_{22}x_{33}x_{44}$ $x_{22}x_{12}x_{33}x_{44}$, 1 3 $+x_{13}(+x_{22}x_{33}x_{44}+x_{42}x_{23}x_{34}+x_{32}x_{43}x_{24}$ $+x_{13}(-x_{22}x_{43}x_{34}-x_{42}x_{33}x_{24}-x_{32}x_{23}x_{44}$ $-x_{23}(-x_{32}x_{13}x_{44}-x_{12}x_{43}x_{34}-x_{42}x_{33}x_{14}$ $-x_{23}(+x_{32}x_{43}x_{14}+x_{12}x_{33}x_{44}+x_{42}x_{13}x_{34}$ $+x_{33}(-x_{42}x_{23}x_{14}-x_{22}x_{13}x_{44}-x_{12}x_{43}x_{24}$ $+x_{33}(+x_{42}x_{13}x_{24}+x_{22}x_{43}x_{14}+x_{12}x_{23}x_{44}$ $-x_{43}(+x_{12}x_{23}x_{34}+x_{32}x_{13}x_{24}+x_{22}x_{33}x_{14}$ $-x_{43}(-x_{12}x_{33}x_{24}-x_{32}x_{23}x_{14}-x_{22}x_{13}x_{34}$ $x_{13}x_{22^{x}33^{x}44}$ $x_{33}x_{22^{x}13^{x}44}$ 1 4 $+x_{14}(+x_{22}x_{33}x_{44}+\overline{x_{42}x_{23}x_{34}}+x_{32}x_{43}x_{24}$ $+x_{14}(-x_{22}x_{43}x_{34}-x_{42}x_{33}x_{24}-x_{32}x_{23}x_{44}$ $-x_{24}(-x_{32}x_{13}x_{44}-x_{12}x_{43}x_{34}-x_{42}x_{33}x_{14}$ $-x_{24}(+x_{32}x_{43}x_{14}+x_{12}x_{33}x_{44}+x_{42}x_{13}x_{34}$ $+x_{34}(-\overline{x_{42}x_{23}x_{14}}-x_{22}x_{13}x_{44}-x_{12}x_{43}x_{24}$ $+x_{34}(+x_{42}x_{13}x_{24}+x_{22}x_{43}x_{14}+x_{12}x_{23}x_{44}$
$x_{11}\cross \begin{array}{llll}x_{25} x_{24} x_{23} x_{22}x_{35} x_{34} x_{33} x_{32}x_{45} x_{44} x_{43} x_{42}x_{55} x_{54} x_{53} x_{52}\end{array} $ $\{\begin{array}{l}x_{11}+x_{12}y+x_{13}y^{2}+x_{14}y^{3}+x_{15}y^{4}=0\ldots(1x_{21}+x_{22}y+x_{23}y^{2}+x_{24}y^{3}+x_{25}y^{4}=0 (2x_{31}+x_{32}y+x_{33}y^{2}+x_{34}y^{3}+x_{35}y^{4}=0\ldots(3x_{41}+x_{42}y+x_{43}y^{2}+x_{44}y^{3}+x_{45}y^{4}=0\ldots(4x_{51}+x_{52}y+x_{53}y^{2}+x_{54}y^{3}+x_{55}y^{4}=0 (5\end{array}$ $x_{53}x_{43}x_{23}x13$ $x_{54}x_{44}x_{24}x14$ $x_{55}x_{45}x_{25}x15$ $x_{43}x_{33}x_{23}x13$ $x_{34}x_{44}xx2414$ $x_{35}xxx_{45}2515$ $=0$ 35 $-x_{44}(+x_{12^{x}23^{x}34}+x_{32^{x}13^{x}24}+x_{22^{x}33^{x}14}$ $-x_{44}(-x_{12}x_{33}x_{24}-x_{32^{x}23^{x}14}-x_{22^{x}13^{x}34}$ $x_{14^{x}22^{x}33^{x}44}$ $x_{44}x_{22}x_{33}x_{14}$ $x_{14^{x}42^{x}23^{x}34}$ $x_{34^{x}42^{x}23^{x}14}$ $U1,$ $D1,$ $U2$ $D2$, ; $U3,$ $D3$ 4 5 12 (12345 (15432 (2345 (5432 6 $x_{22}$ $x_{23}$ $x_{24}$ $x_{25}$ $x_{11}\cross$ $x_{32}$ $x_{33}$ $x_{34}$ $x_{3_{d}^{r}}$ $x_{42}$ $x_{43}$ $x_{44}$ $x_{45}$ $x_{52}$ $x_{53}$ $x_{54}$ $x_{55}$ (5 120 O 60 2 12 120 5 0 $y$ 5 4 5 (2,(3,(4, (5 4 $y,$ $y^{2},$ $y^{3},$ $y^{4}$ $y$ $0$ 1 2 3 4 4 5 (1, (2,(3,(4, (5 $x_{22}$ $x_{23}$ $x_{24}$ $x_{25}$ $x_{32}$ $x_{33}$ $x_{34}$ $x_{35}$ $x_{11}\cross$ $+$ $x_{21}\cross(-1 \begin{array}{llll}x_{12} x_{13} x_{14} x_{1_{\mathfrak{d}}^{r}}x_{32} x_{33} x_{34} x_{35}x_{42} x_{43} x_{44} x_{45}x_{52} x_{53} x_{54} x_{55}\end{array} +x_{31}\cross x_{52}x_{42}xx2212$ $x_{42}$ $x_{43}$ $x_{44}$ $x_{45}$ $x_{52}$ $x_{53}$ $x_{54}$ $x_{55}$ $+$ $x_{41}\cross(-1 \begin{array}{llll}x_{12} x_{13} x_{14} x_{15}x_{22} x_{23} x_{24} x_{25}x_{32} x_{33} x_{34} x_{35}x_{52} x_{53} x_{\iota}r_{}4 x_{55}\end{array} +x_{51}\cross x_{32}x_{42}xx22$
36 [1] Goto, T and Komatsu, H: Determinants, resultants and discriminants in Japan in the seventeenth century and in Europe in eighteenth and nineteenth centuries, J Northwest University (Natural Science Edition, 33 No3, pp 363-367 (2003 [2] : (1974 [3] 104 3 pp313-333