余因子行列
\(i\)行\(j\)列の要素を\(a_{ij}\)とする\(n\)次正方行列\(\boldsymbol{A}\)において，
\(i\)行と\(j\)列の要素を取り除いた残りの行列を\(\boldsymbol{M}_{ij}\)とした時，
\(\tilde{a}_{ij}=(-1)^{i+j}\left|\boldsymbol{M}_{ji}\right|(添え字の順序に注意)\)を要素とする行列\(\tilde{\boldsymbol{A}}\)を余因子行列と呼ぶ．
2x2行列
$$\begin{eqnarray}
\boldsymbol{A}_{2\times2}&=&
\begin{bmatrix}
a_{11} & a_{12}
\\ a_{21} & a_{22}
\end{bmatrix}
\\
\\\tilde{\boldsymbol{A}}_{2\times2}&=&
\begin{bmatrix}
(-1)^{1+1}\left|\boldsymbol{M}_{11}\right| & (-1)^{1+2}\left|\boldsymbol{M}_{21}\right|
\\(-1)^{2+1}\left|\boldsymbol{M}_{12}\right| & (-1)^{2+2}\left|\boldsymbol{M}_{22}\right|
\end{bmatrix}
\\&=&
\begin{bmatrix}
\left|\boldsymbol{M}_{11}\right| & -\left|\boldsymbol{M}_{21}\right|
\\-\left|\boldsymbol{M}_{12}\right| & \left|\boldsymbol{M}_{22}\right|
\end{bmatrix}
\\&=&
\begin{bmatrix}
\left|\boldsymbol{M}_{11}\right| & -\left|\boldsymbol{M}_{12}\right|
\\-\left|\boldsymbol{M}_{21}\right| & \left|\boldsymbol{M}_{22}\right|
\end{bmatrix}^{T}\;\ldots\;\boldsymbol{A}^{T}:\boldsymbol{A}の転置行列
\\&=&
\begin{bmatrix}
a_{22} & -a_{21}
\\ -a_{12} & a_{11}
\end{bmatrix}^{T}
\\&=&
\begin{bmatrix}
a_{22} & -a_{12}
\\ -a_{21} & a_{11}
\end{bmatrix}
\\
\\\boldsymbol{A}_{2\times2}&=&
\begin{bmatrix}
a& b
\\c & d
\end{bmatrix}
\\\tilde{\boldsymbol{A}}_{2\times2}&=&
\begin{bmatrix}
d & -b
\\ -c & a
\end{bmatrix}
\end{eqnarray}$$
3x3行列
$$\begin{eqnarray}
\boldsymbol{A}_{3\times3}&=&
\begin{bmatrix}
a_{11} & a_{12} & a_{13}
\\ a_{21} & a_{22} & a_{23}
\\ a_{31} & a_{32} & a_{33}
\end{bmatrix}
\\
\\\tilde{\boldsymbol{A}}_{3\times3}
&=&
\begin{bmatrix}
(-1)^{1+1}\left|\boldsymbol{M}_{11}\right| & (-1)^{1+2}\left|\boldsymbol{M}_{21}\right| & (-1)^{1+3}\left|\boldsymbol{M}_{31}\right|
\\(-1)^{2+1}\left|\boldsymbol{M}_{12}\right| & (-1)^{2+2}\left|\boldsymbol{M}_{22}\right| & (-1)^{2+3}\left|\boldsymbol{M}_{32}\right|
\\(-1)^{3+1}\left|\boldsymbol{M}_{13}\right| & (-1)^{3+2}\left|\boldsymbol{M}_{23}\right| & (-1)^{3+3}\left|\boldsymbol{M}_{33}\right|
\end{bmatrix}
\\&=&
\begin{bmatrix}
\left|\boldsymbol{M}_{11}\right| & -\left|\boldsymbol{M}_{21}\right| & \left|\boldsymbol{M}_{31}\right|
\\-\left|\boldsymbol{M}_{12}\right| & \left|\boldsymbol{M}_{22}\right| & -\left|\boldsymbol{M}_{32}\right|
\\\left|\boldsymbol{M}_{13}\right| & -\left|\boldsymbol{M}_{23}\right| & \left|\boldsymbol{M}_{33}\right|
\end{bmatrix}
\\&=&
\begin{bmatrix}
\left|\boldsymbol{M}_{11}\right| & -\left|\boldsymbol{M}_{12}\right| & \left|\boldsymbol{M}_{13}\right|
\\-\left|\boldsymbol{M}_{21}\right| & \left|\boldsymbol{M}_{22}\right| & -\left|\boldsymbol{M}_{23}\right|
\\\left|\boldsymbol{M}_{31}\right| & -\left|\boldsymbol{M}_{32}\right| & \left|\boldsymbol{M}_{33}\right|
\end{bmatrix}^{T}\;\ldots\;\boldsymbol{A}^{T}:\boldsymbol{A}の転置行列
\\&=&
\begin{bmatrix}
\begin{vmatrix}
a_{22} & a_{23}
\\ a_{32} & a_{33}
\end{vmatrix}
& -\begin{vmatrix}
a_{21} & a_{23}
\\ a_{31} & a_{33}
\end{vmatrix}
& \begin{vmatrix}
a_{21} & a_{22}
\\ a_{31} & a_{32}
\end{vmatrix}
\\-\begin{vmatrix}
a_{12} & a_{13}
\\ a_{32} & a_{33}
\end{vmatrix}
& \begin{vmatrix}
a_{11} & a_{13}
\\ a_{31} & a_{33}
\end{vmatrix}
& -\begin{vmatrix}
a_{11} & a_{12}
\\ a_{31} & a_{32}
\end{vmatrix}
\\\begin{vmatrix}
a_{12} & a_{13}
\\ a_{22} & a_{23}
\end{vmatrix}
& -\begin{vmatrix}
a_{11} & a_{13}
\\ a_{21} & a_{23}
\end{vmatrix}
& \begin{vmatrix}
a_{11} & a_{12}
\\ a_{21} & a_{22}
\end{vmatrix}
\end{bmatrix}^{T}
\\&=&
\begin{bmatrix}
a_{22}a_{33}-a_{23}a_{32}
& -(a_{21}a_{33}-a_{23}a_{31})
& a_{21}a_{32}-a_{22}a_{31}
\\-(a_{12}a_{33}-a_{13}a_{32})
& a_{11}a_{33}-a_{13}a_{31}
& -(a_{11}a_{32}-a_{12}a_{31})
\\a_{12}a_{23}-a_{13}a_{22}
& -(a_{11}a_{23}-a_{13}a_{21})
& a_{11}a_{22}-a_{12}a_{21}
\end{bmatrix}^{T}
\\&=&
\begin{bmatrix}
a_{22}a_{33}-a_{23}a_{32} & -(a_{12}a_{33}-a_{13}a_{32}) & a_{12}a_{23}-a_{13}a_{22}
\\-(a_{21}a_{33}-a_{23}a_{31}) & a_{11}a_{33}-a_{13}a_{31} & -(a_{11}a_{23}-a_{13}a_{21})
\\a_{21}a_{32}-a_{22}a_{31} & -(a_{11}a_{32}-a_{12}a_{31}) & a_{11}a_{22}-a_{12}a_{21}
\end{bmatrix}
\\&=&
\begin{bmatrix}
a_{22}a_{33}-a_{23}a_{32} & a_{13}a_{32}-a_{12}a_{33} & a_{12}a_{23}-a_{13}a_{22}
\\a_{23}a_{31}-a_{21}a_{33} & a_{11}a_{33}-a_{13}a_{31} & a_{13}a_{21}-a_{11}a_{23}
\\a_{21}a_{32}-a_{22}a_{31} & a_{12}a_{31}-a_{11}a_{32} & a_{11}a_{22}-a_{12}a_{21}
\end{bmatrix}
\end{eqnarray}$$
余因子行列と元の行列との積
$$\begin{eqnarray}
\tilde{\boldsymbol{A}}\boldsymbol{A}=\boldsymbol{A}\tilde{\boldsymbol{A}}=\left|\boldsymbol{A}\right|\boldsymbol{I}
\end{eqnarray}$$
余因子行列と元の行列との積 3x3の例
$$\begin{eqnarray}
\tilde{\boldsymbol{A}}_{3\times3}\boldsymbol{A}_{3\times3}
&=&
\begin{bmatrix}
\left|\boldsymbol{M}_{11}\right| & -\left|\boldsymbol{M}_{21}\right| & \left|\boldsymbol{M}_{31}\right|
\\-\left|\boldsymbol{M}_{12}\right| & \left|\boldsymbol{M}_{22}\right| & -\left|\boldsymbol{M}_{32}\right|
\\\left|\boldsymbol{M}_{13}\right| & -\left|\boldsymbol{M}_{23}\right| & \left|\boldsymbol{M}_{33}\right|
\end{bmatrix}
\begin{bmatrix}
a_{11} & a_{12} & a_{13}
\\ a_{21} & a_{22} & a_{23}
\\ a_{31} & a_{32} & a_{33}
\end{bmatrix}
\\\\&=&\begin{bmatrix}
\left|\boldsymbol{M}_{11}\right| a_{11} -\left|\boldsymbol{M}_{21}\right| a_{21} +\left|\boldsymbol{M}_{31}\right| a_{31}
& \left|\boldsymbol{M}_{11}\right| a_{12} -\left|\boldsymbol{M}_{21}\right| a_{22} +\left|\boldsymbol{M}_{31}\right| a_{32}
& \left|\boldsymbol{M}_{11}\right| a_{13} -\left|\boldsymbol{M}_{21}\right| a_{23} +\left|\boldsymbol{M}_{31}\right| a_{33}
\\-\left|\boldsymbol{M}_{12}\right| a_{11} +\left|\boldsymbol{M}_{22}\right| a_{21} -\left|\boldsymbol{M}_{32}\right| a_{31}
& -\left|\boldsymbol{M}_{12}\right| a_{12} +\left|\boldsymbol{M}_{22}\right| a_{22} -\left|\boldsymbol{M}_{32}\right| a_{32}
& -\left|\boldsymbol{M}_{12}\right| a_{13} +\left|\boldsymbol{M}_{22}\right| a_{23} -\left|\boldsymbol{M}_{32}\right| a_{33}
\\\left|\boldsymbol{M}_{13}\right| a_{11} -\left|\boldsymbol{M}_{23}\right| a_{21} +\left|\boldsymbol{M}_{33}\right| a_{31}
& \left|\boldsymbol{M}_{13}\right| a_{12} -\left|\boldsymbol{M}_{23}\right| a_{22} +\left|\boldsymbol{M}_{33}\right| a_{32}
& \left|\boldsymbol{M}_{13}\right| a_{13} -\left|\boldsymbol{M}_{23}\right| a_{23} +\left|\boldsymbol{M}_{33}\right| a_{33}
\end{bmatrix}
\\\\&=&\begin{bmatrix}
\left|\boldsymbol{A}_{3\times3}\right|
& \left|\boldsymbol{M}_{11}\right| a_{12} -\left|\boldsymbol{M}_{21}\right| a_{22} +\left|\boldsymbol{M}_{31}\right| a_{32}
& \left|\boldsymbol{M}_{11}\right| a_{13} -\left|\boldsymbol{M}_{21}\right| a_{23} +\left|\boldsymbol{M}_{31}\right| a_{33}
\\-\left|\boldsymbol{M}_{12}\right| a_{11} +\left|\boldsymbol{M}_{22}\right| a_{21} -\left|\boldsymbol{M}_{32}\right| a_{31}
& \left|\boldsymbol{A}_{3\times3}\right|
& -\left|\boldsymbol{M}_{12}\right| a_{13} +\left|\boldsymbol{M}_{22}\right| a_{23} -\left|\boldsymbol{M}_{32}\right| a_{33}
\\\left|\boldsymbol{M}_{13}\right| a_{11} -\left|\boldsymbol{M}_{23}\right| a_{21} +\left|\boldsymbol{M}_{33}\right| a_{31}
& \left|\boldsymbol{M}_{13}\right| a_{12} -\left|\boldsymbol{M}_{23}\right| a_{22} +\left|\boldsymbol{M}_{33}\right| a_{32}
& \left|\boldsymbol{A}_{3\times3}\right|
\end{bmatrix}
\;\ldots\;計算は下記“対角要素について”に記載
\\\\&=&\begin{bmatrix}
\left|\boldsymbol{A}_{3\times3}\right| & 0 & 0
\\0 & \left|\boldsymbol{A}_{3\times3}\right| & 0
\\0 & 0 & \left|\boldsymbol{A}_{3\times3}\right|
\end{bmatrix}
\;\ldots\;計算は下記“その他要素について”に記載
\\\\&=&\left|\boldsymbol{A}_{3\times3}\right|\begin{bmatrix}
1 & 0 & 0
\\0 & 1 & 0
\\0 & 0 & 1
\end{bmatrix}
\\\\&=&\left|\boldsymbol{A}_{3\times3}\right| \boldsymbol{I}_{3}
\;\ldots\;\boldsymbol{I}_{3}:3\times3の単位行列
\end{eqnarray}$$
対角要素について
$$\begin{eqnarray}
\left|\boldsymbol{A}_{3\times3}\right|&=&
\begin{vmatrix}
a_{11} & a_{12} & a_{13}
\\ a_{21} & a_{22} & a_{23}
\\ a_{31} & a_{32} & a_{33}
\end{vmatrix}
\\&=&a_{11}\boldsymbol{M}_{11}
-a_{21}\boldsymbol{M}_{21}
+a_{31}\boldsymbol{M}_{31}
\\&=&-a_{12}\boldsymbol{M}_{12}
+a_{22}\boldsymbol{M}_{22}
-a_{32}\boldsymbol{M}_{32}
\\&=&a_{13}\boldsymbol{M}_{13}
-a_{23}\boldsymbol{M}_{23}
+a_{33}\boldsymbol{M}_{33}
\end{eqnarray}$$
その他要素について
$$\begin{eqnarray}
\left|\boldsymbol{M}_{11}\right| a_{12} -\left|\boldsymbol{M}_{21}\right| a_{22} +\left|\boldsymbol{M}_{31}\right| a_{32}
&=&
\begin{vmatrix}
a_{22} & a_{23}
\\ a_{32} & a_{33}
\end{vmatrix} a_{12}
-\begin{vmatrix}
a_{12} & a_{13}
\\ a_{32} & a_{33}
\end{vmatrix} a_{22}
+\begin{vmatrix}
a_{12} & a_{13}
\\ a_{22} & a_{23}
\end{vmatrix} a_{32}
\\&=&a_{12}(a_{22}a_{33}-a_{23}a_{32})-a_{22}(a_{12}a_{33}-a_{13}a_{32})+a_{32}(a_{12}a_{23}-a_{13}a_{22})
\\&=&
\color{red }{ a_{12}a_{22}a_{33}}
\color{blue }{-a_{12}a_{23}a_{32}}
\color{red }{-a_{12}a_{22}a_{33}}
\color{green}{+a_{13}a_{22}a_{32}}
\color{blue }{+a_{12}a_{23}a_{32}}
\color{green}{-a_{13}a_{22}a_{32}}
\\&=&0
\\\left|\boldsymbol{M}_{11}\right| a_{13} -\left|\boldsymbol{M}_{21}\right| a_{23} +\left|\boldsymbol{M}_{31}\right| a_{33}
&=&
\begin{vmatrix}
a_{22} & a_{23}
\\ a_{32} & a_{33}
\end{vmatrix} a_{13}
-\begin{vmatrix}
a_{12} & a_{13}
\\ a_{32} & a_{33}
\end{vmatrix} a_{23}
+\begin{vmatrix}
a_{12} & a_{13}
\\ a_{22} & a_{23}
\end{vmatrix} a_{33}
\\&=&a_{13}(a_{22}a_{33}-a_{23}a_{32})-a_{23}(a_{12}a_{33}-a_{13}a_{32})+a_{33}(a_{12}a_{23}-a_{13}a_{22})
\\&=&
\color{red }{ a_{13}a_{22}a_{33}}
\color{blue }{-a_{13}a_{23}a_{32}}
\color{green}{-a_{12}a_{23}a_{33}}
\color{blue}{+a_{13}a_{23}a_{32}}
\color{green}{+a_{12}a_{23}a_{33}}
\color{red }{-a_{13}a_{22}a_{33}}
\\&=&0
\\-\left|\boldsymbol{M}_{12}\right| a_{11} +\left|\boldsymbol{M}_{22}\right| a_{21} -\left|\boldsymbol{M}_{32}\right| a_{31}
&=&
-\begin{vmatrix}
a_{21} & a_{23}
\\ a_{31} & a_{33}
\end{vmatrix} a_{11}
+\begin{vmatrix}
a_{11} & a_{13}
\\ a_{31} & a_{33}
\end{vmatrix} a_{21}
-\begin{vmatrix}
a_{11} & a_{13}
\\ a_{21} & a_{23}
\end{vmatrix} a_{31}
\\&=&-a_{11}(a_{21}a_{33}-a_{23}a_{31})+a_{21}(a_{11}a_{33}-a_{13}a_{31})-a_{31}(a_{11}a_{23}-a_{13}a_{21})
\\&=&
\color{red }{-a_{11}a_{21}a_{33}}
\color{blue }{+a_{11}a_{23}a_{31}}
\color{red }{+a_{11}a_{21}a_{33}}
\color{green}{-a_{13}a_{21}a_{31}}
\color{blue }{-a_{11}a_{23}a_{31}}
\color{green}{+a_{13}a_{21}a_{31}}
\\&=&0
\\-\left|\boldsymbol{M}_{12}\right| a_{13} +\left|\boldsymbol{M}_{22}\right| a_{23} -\left|\boldsymbol{M}_{32}\right| a_{33}
&=&
-\begin{vmatrix}
a_{21} & a_{23}
\\ a_{31} & a_{33}
\end{vmatrix} a_{13}
+\begin{vmatrix}
a_{11} & a_{13}
\\ a_{31} & a_{33}
\end{vmatrix} a_{23}
-\begin{vmatrix}
a_{11} & a_{13}
\\ a_{21} & a_{23}
\end{vmatrix} a_{33}
\\&=&-a_{13}(a_{21}a_{33}-a_{23}a_{31})+a_{23}(a_{11}a_{33}-a_{13}a_{31})-a_{33}(a_{11}a_{23}-a_{13}a_{21})
\\&=&
\color{red }{-a_{13}a_{21}a_{33}}
\color{blue }{+a_{13}a_{23}a_{31}}
\color{green}{+a_{11}a_{23}a_{33}}
\color{blue }{-a_{13}a_{23}a_{31}}
\color{green}{-a_{11}a_{23}a_{33}}
\color{red }{+a_{13}a_{21}a_{33}}
\\&=&0
\\
\left|\boldsymbol{M}_{13}\right| a_{11}
-\left|\boldsymbol{M}_{23}\right| a_{21}
+\left|\boldsymbol{M}_{33}\right| a_{31}
&=&
\begin{vmatrix}
a_{21} & a_{22}
\\ a_{31} & a_{32}
\end{vmatrix} a_{11}
-\begin{vmatrix}
a_{11} & a_{12}
\\ a_{31} & a_{32}
\end{vmatrix} a_{21}
+\begin{vmatrix}
a_{11} & a_{12}
\\ a_{21} & a_{22}
\end{vmatrix} a_{31}
\\&=&a_{11}(a_{21}a_{32}-a_{22}a_{31})
-a_{21}(a_{11}a_{32}-a_{12}a_{31})
+a_{31}(a_{11}a_{22}-a_{12}a_{21})
\\&=&
\color{red }{ a_{11}a_{21}a_{32}}
\color{blue }{-a_{11}a_{22}a_{31}}
\color{red }{-a_{11}a_{21}a_{32}}
\color{green}{+a_{12}a_{21}a_{31}}
\color{blue }{+a_{11}a_{22}a_{31}}
\color{green}{-a_{12}a_{21}a_{31}}
\\&=&0
\\
\left|\boldsymbol{M}_{13}\right| a_{12}
-\left|\boldsymbol{M}_{23}\right| a_{22}
+\left|\boldsymbol{M}_{33}\right| a_{32}
&=&
\begin{vmatrix}
a_{21} & a_{22}
\\ a_{31} & a_{32}
\end{vmatrix} a_{12}
-\begin{vmatrix}
a_{11} & a_{12}
\\ a_{31} & a_{32}
\end{vmatrix} a_{22}
+\begin{vmatrix}
a_{11} & a_{12}
\\ a_{21} & a_{22}
\end{vmatrix} a_{32}
\\&=&
a_{12}(a_{21}a_{32}-a_{22}a_{31})
-a_{22}(a_{11}a_{32}-a_{12}a_{31})
+a_{32}(a_{11}a_{22}-a_{12}a_{21})
\\&=&
\color{red }{ a_{12}a_{21}a_{32}}
\color{blue }{-a_{12}a_{22}a_{31}}
\color{green}{-a_{11}a_{22}a_{32}}
\color{blue}{+a_{12}a_{22}a_{31}}
\color{green}{+a_{11}a_{22}a_{32}}
\color{red }{-a_{12}a_{21}a_{32}}
\\&=&0
\end{eqnarray}$$
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