\(\textsf{行列式の展開とは、$n$次正方行列(次数$n$)の行列式を$n\!-\!{\small 1}$次正方行列(次数$n\!-\!{\small 1}$)の行列式(小行列)の積の総和の形式に展開することである。}\)
余因子
$$detA=\begin{vmatrix}{\large a}{\tiny 11}&{\large a}{\tiny 12}&{\large a}{\tiny 13} \\ {\large a}{\tiny 21}&{\large a}{\tiny 22}&{\large a}{\tiny 23} \\ {\large a}{\tiny 31}&{\large a}{\tiny 32}&{\large a}{\tiny 33}\end{vmatrix}
={\large a}{\tiny 11}\overbrace{\begin{vmatrix}{\large a}{\tiny 22}&{\large a}{\tiny 23} \\ {\large a}{\tiny 32}&{\large a}{\tiny 33}\end{vmatrix}}^{{\Large A}{\tiny 1\,1}}{\,\small -\,}{\large a}{\tiny 12}\overbrace{\begin{vmatrix}{\large a}{\tiny 21}&{\large a}{\tiny 23} \\ {\large a}{\tiny 31}&{\large a}{\tiny 33}\end{vmatrix}}^{{\Large A}{\tiny 1\,2}}{\,\small +\,}{\large a}{\tiny 13}\overbrace{\begin{vmatrix}{\large a}{\tiny 21}&{\large a}{\tiny 22} \\ {\large a}{\tiny 31}&{\large a}{\tiny 32}\end{vmatrix}}^{{\Large A}{\tiny 1\,3}}$$
\(\textsf{それぞれの小行列の上部に表記された$A{\tiny i\,j}$ を行列$A$の成分${\large a}{\tiny i\,j}$ の 余因子 という。}\)
\(\textsf{小行列を$M{\tiny i\,j}$ 、行列式を$|M{\tiny i\,j}|$ と表記し、余因子を $A{\tiny i\,j}={\large\tilde a}{\tiny i\,j}=(-1)^{\scriptsize i{\tiny +\,}j}|M{\tiny i\,j}|$と定義する。}\)
\(\textsf{余因子${\large a}{\tiny i\,j}$ の事例}\left\{\begin{array}{l}\textsf{任意の一行一列を取り除く
(ここでは$\color{red}2$行$\color{red}2$列とする)。} \\[10pt] \underset{\overset{\phantom{a}}{}}{\begin{vmatrix}{\large a\tiny 11}&\color{red}{\large a\tiny 12}&{\large a\tiny 13} \\ \color{red}
{\large a\tiny 21}&\color{red}{\large a\tiny 22}&\color{red}{\large a\tiny 23} \\{\large a\tiny 31}&\color{red}{\large a\tiny 32}&{\large a\tiny 33}\end{vmatrix}}\hspace{10pt}\Rightarrow\hspace{10pt}\underset{\overset{\phantom{\large a}}{\huge\tilde a \scriptsize \color{red} 22}}
{\begin{vmatrix}{\large a\tiny 11}&{\large a\tiny 13} \\
{\large a\tiny 31}&{\large a\tiny 33}\end{vmatrix}} \\[30pt] \textsf{小行列の行列式($M\tiny i\,j$)をもとめる。}M{\tiny\color{red}2\,2}=\large a
{\tiny{11}}\,\large a{\tiny{33}}\:{\scriptsize -}\,\large a{\tiny{13}}\,\large a{\tiny{31}} \\[30pt]
\textsf{余因子$\large\tilde{a}\tiny i\,j$の値}(\textsf{$\large\tilde a \color{red}\tiny 22$})=(-1)^{i+j}({\small =(-1)^{2+2}})\times M{\color{red}\tiny 2\,2}=\large a{\tiny{11}}\,\large a{\tiny{33}}\:{\scriptsize -}\,\large a{\tiny{13}}\,\large a{\tiny{31}}
\end{array}\right\}\)
行列式の余因子展開
\(\begin{array}{l}\textsf{余因子展開とは、$n$次正方行列の任意の行(第$i$行$\small=1,\,2,\,\cdots\, ,n$)とその成分${\large a}{\tiny i\,j}({\small j=1,\,2,\,\cdots,\,n})$} \\[5pt]\hspace{10pt}\textsf{ または任意の列(第$j$列$\small=1,\,2,\,\cdots\, ,n$)とその成分${\large a}{\tiny i\,j}({\small i=1,\,2,\,\cdots,\,n})$に対応する余因子${\large\tilde a}{\tiny i\,j}$との積の総和による展開式である。}\end{array}\)
$\begin{eqnarray}det(A)=\begin{vmatrix}{\large}a{\tiny 11} & {\large}a{\tiny 12} & \ldots & {\large}a{\tiny 1n} \\ {\large}a{\tiny 21} & {\large}a{\tiny 22} & \ldots & {\large}a{\tiny 2n}\\ \vdots & \vdots & \ddots & \vdots \\{\large}a{\tiny n1} & {\large}a{\tiny n2} & \ldots & {\large}a{\tiny nn}\end{vmatrix}\end{eqnarray}$
$$\bbox[#f5f5f5, 3pt, border:1px solid]{\class{Boldfont}{\:\textsf{余因子展開の定理}}\hspace{10pt}det(A){\:\scriptsize =\:}{\left\{\begin{array}{l}\small\textsf{第$i$行による余因子
展開}\hspace{10pt}\overset{\:\:\small n}{\underset{\tiny J\:=\:1}{\LARGE\varSigma}}{\large a\tiny i\,J}\:{\large \tilde{a}\tiny i\,J}\hspace{5pt}({\scriptsize =}\:{\large a\tiny i\,1}\,
{\large \tilde{a}\tiny i\,1}{\:\scriptsize +\:}{\large a\tiny i\,2}\,{\large \tilde{a}\tiny i\,2}{\:\scriptsize +\:}{\large a\tiny i\,3}\,{\large \tilde{a}\tiny i\,3}{\:\scriptsize +\:}
\cdots{\:\scriptsize +\:}{\large a\tiny i\,n}\,{\large \tilde{a}\tiny i\,n})\\ \hspace{150pt}\scriptsize{\class{Boldfont}{\vert\vert}} \\ \small\textsf{第$j$列による余因子展開}\hspace{10pt}
\overset{\:\:\small n}{\underset{\tiny I\:=\:1}{\LARGE\varSigma}}{\large a\tiny I\,j}\:{\large \tilde{a}\tiny I,j}\hspace{5pt}({\scriptsize =}\:{\large a\tiny 1\,j}\,
{\large \tilde{a}\tiny 1\,j}{\:\scriptsize +\:}{\large a\tiny 2\,j}\,{\large \tilde{a}\tiny 2\,j}{\:\scriptsize +\:}{\large a\tiny 3\,j}\,{\large \tilde{a}\tiny 3\,j}{\:\scriptsize +\:}\cdots
{\:\scriptsize +\:}{\large a\tiny n\,j}\,{\large \tilde{a}\tiny n\,j})\end{array}\right\}}}$$
\(2\times2\) 行列式の余因子展開
$$\hspace{-20pt}det(A)=\begin{vmatrix}{\large a}{\tiny 11}&{\large a}{\tiny 12} \\ {\large a}{\tiny 21}&{\large a}{\tiny 22}\end{vmatrix}=\begin{cases}\overset{\:\:\small n}{\underset{\tiny J\:=\:1}{\LARGE\varSigma}}{\large a\tiny i\,J}\:{\large \tilde{a}\tiny i\,J}\hspace{5pt}({\scriptsize =}\:{\large a\tiny i\,1}\,
{\large \tilde{a}\tiny i\,1}{\:\scriptsize +\:}{\large a\tiny i\,2}\,{\large \tilde{a}\tiny i\,2})\begin{cases}\Rightarrow\small\textsf{第$1$行の成分$\large a\tiny{\color{red}1}\,J$}\begin{cases}{\large a}{\tiny {\color{red}1}1} \\ {\phantom a}\hspace{10pt}\begin{vmatrix}\color{red}{\large a}{\tiny11}&\color{red}{\large a}{\tiny12} \\ {\large a}{\tiny 21}&{\large a}{\tiny 22}\end{vmatrix}\hspace{10pt}\textsf{第$1$行の余因子}\begin{cases}{\large \tilde{a}\tiny{\color{red}1}1}{\,\scriptsize =\,}(\small -1)^{\tiny{\color{red}1}\,+\,1}\times {M\tiny{\color{red}1}1}{\,\scriptsize =\,}\hspace{8pt}{\small 1}\times\begin{vmatrix}1&0 \\ 0&a{\tiny 22}\end{vmatrix}{\,\scriptsize =\,}\hspace{8pt}{\large a}{\tiny 22} \\[5pt] {\large \tilde{a}\tiny{\color{red}1}2}{\,\scriptsize =\,}(\small -1)^{\tiny{\color{red}1}\,+\,2}\times{M\tiny{\color{red}1}2}{\,\scriptsize =\,}{\small -1}\times\begin{vmatrix}0&1 \\ a{\tiny 21}&0\end{vmatrix}{\,\scriptsize =\,}{\,\scriptsize -}{\large a}{\tiny 21}\end{cases} \\ {\large a}{\tiny{\color{red}1}2}\end{cases} \\[10pt] {\,\scriptsize =\,}\enclose{roundedbox}{{\large a}{\tiny 11}{\large a}{\tiny 22}{\,\scriptsize -\,}{\large a}{\tiny 12}{\large a}{\tiny 21}}\end{cases} \\[30pt] \overset{\:\:\small n}{\underset{\tiny I\:=\:1}{\LARGE\varSigma}}{\large a\tiny I\,j}\:{\large \tilde{a}\tiny I\,j}\hspace{5pt}({\scriptsize =}\:{\large a\tiny 1\,j}\,
{\large \tilde{a}\tiny 1\,j}{\:\scriptsize +\:}{\large a\tiny 2\,j}\,{\large \tilde{a}\tiny 2\,j})\begin{cases}\Rightarrow\small\textsf{第$1$列の成分$\large a\tiny 1\,{\color{red}1}$}\begin{cases}{\large a}{\tiny 1{\color{red}1}} \\ {\phantom a}\hspace{10pt}\begin{vmatrix}\color{red}{\large a}{\tiny11}&{\large a}{\tiny12} \\ \color{red}{\large a}{\tiny 21}&{\large a}{\tiny 22}\end{vmatrix}\hspace{10pt}\textsf{第$1$列の余因子}\begin{cases}{\large \tilde{a}\tiny 1{\color{red}1}}{\,\scriptsize =\,}(\small -1)^{\tiny 1\,+\,{\color{red}1}}\times {M\tiny 1{\color{red}1}}{\,\scriptsize =\,}\hspace{8pt}{\small 1}\times\begin{vmatrix}1&0 \\ 0&a{\tiny 22}\end{vmatrix}{\,\scriptsize =\,}\hspace{8pt}{\large a}{\tiny 22} \\ {\large \tilde{a}\tiny 2{\color{red}1}}{\,\scriptsize =\,}(\small -1)^{\tiny 2\,+\,{\color{red}1}}\times{M\tiny 2{\color{red}1}}{\,\scriptsize =\,}{\small -1}\times\begin{vmatrix}0&a{\tiny 12} \\ 1&0\end{vmatrix}{\,\scriptsize =\,}{\,\scriptsize -}{\large a}{\tiny 21}\end{cases} \\ {\large a}{\tiny 2{\color{red}1}}\end{cases} \\[10pt] {\,\scriptsize =\,}\enclose{roundedbox}{{\large a}{\tiny 11}{\large a}{\tiny 22}{\,\scriptsize -\,}{\large a}{\tiny 12}{\large a}{\tiny 21}}\end{cases}\end{cases}$$
\(\textsf{プレート$\,\enclose{roundedbox}{\scriptsize\phantom{abcde}}\,$内が解である。以降も同様。}\)
\(3\times3\) 行列式の余因子展開
$$\hspace{-20pt}{detA=\begin{vmatrix}{a}{\tiny 11}&{a}{\tiny 12}&{a}{\tiny 13} \\ {a}{\tiny 21}&{a}{\tiny 22}&{a}{\tiny 23} \\ {a}{\tiny 31}&{a}{\tiny 32}&{a}{\tiny 33}\end{vmatrix}=\begin{cases}\overset{\:\:\small n}{\underset{\tiny J\:=\:1}{\LARGE\varSigma}}{\large a\tiny i\,J}\:{\large \tilde{a}\tiny i\,J}\hspace{5pt}\left\{\begin{array}{l}\Rightarrow\small\textsf{第$2$行の成分 $\large a\tiny{\color{red}2}\,J$}\left\{\begin{array}{l}{\large a\tiny{\color{red}2}1} \\ {\large a\tiny{\color{red}2}2} \hspace{10pt}\begin{vmatrix}{a\tiny 11}&{a\tiny 12}&{a\tiny 13} \\ \color{red}{a\tiny 21}&\color{red}{a\tiny 22}&\color{red}{a\tiny 23} \\ {a\tiny 31}&{a\tiny 32}&{a\tiny 33}\end{vmatrix} \\ {\large a\tiny{\color{red}2}3}\end{array}\right.\hspace{10pt}\textsf{第$2$行の余因子} \left\{\begin{array}{l}{\large\tilde a\tiny{\color{red} 2}1}{\,\scriptsize =\,}({\scriptsize -}{\small 1})^{\color{red}\tiny 2\color{black}\,+\,1}\,\times{M\tiny{\color{red} 2}1}{\,\scriptsize =\,}{{\scriptsize -}\small 1}\times \begin{vmatrix}a\tiny 12&a\tiny 13 \\ a\tiny 32& a\tiny 33 \end{vmatrix}{\,\scriptsize =\,}{\scriptsize -\,}\large a\tiny 12\large a\tiny 33{\,\scriptsize +\,}\large a\tiny 13\large a\tiny 32 \\ {\large\tilde a\tiny{\color{red} 2}2}{\,\scriptsize =\,}({\scriptsize -}{\small 1})^{\color{red}\tiny 2\color{black}\,+\,2}\,\times{M\tiny{\color{red} 2}2}{\,\scriptsize =\,}\hspace{6pt}{\small 1}\times \begin{vmatrix}a\tiny 11& a\tiny 13 \\ a\tiny 31& a\tiny 33 \end{vmatrix}{\,\scriptsize =\,}\hspace{8pt}\large a\tiny 11 \large a\tiny 33{\,\scriptsize -\,}\large a\tiny 13 \large a\tiny 31 \\ {\large\tilde a\tiny{\color{red} 2}3}{\,\scriptsize =\,}({\scriptsize -}{\small 1})^{\color{red}\tiny 2\color{black}\,+\,3}\,\times{M\tiny{\color{red} 2}3}{\,\scriptsize =\,}{{\scriptsize -}\small 1}\times \begin{vmatrix}a\tiny 11& a\tiny 12 \\ a\tiny 31&a\tiny 32 \end{vmatrix}{\,\scriptsize =\,}{\scriptsize -\,}\large a\tiny 11\large a\tiny 32{\,\scriptsize +\,}\large a\tiny 12\large a\tiny 31 \end{array}\right. \\[10pt] {\,\scriptsize=\,}{\large a\tiny 21}({\scriptsize-\,}{\large a\tiny 12}{\large a\tiny 33}{\,\scriptsize+\,}{\large a\tiny 13}{\large a\tiny 32}){\,\scriptsize+\,}{\large a\tiny 22}({\large a\tiny 11}{\large a\tiny 33}{\,\scriptsize-\,}{\large a\tiny 13}{\large a\tiny 31}){\,\scriptsize+\,}{\large a\tiny 23}({\scriptsize-\,}{\large a\tiny 11}{\large a\tiny 32}{\,\scriptsize+\,}{\large a\tiny 12}{\large a\tiny 31}) \\{\,\scriptsize=\,}{\scriptsize-\,}{\large a\tiny 12}{\large a\tiny 21}{\large a\tiny 33}{\,\scriptsize+\,}{\large a\tiny 13}{\large a\tiny 21}{\large a\tiny 32}{\,\scriptsize+\,}{\large a\tiny 11}{\large a\tiny 22}{\large a\tiny 33}{\,\scriptsize-\,}{\large a\tiny 13}{\large a\tiny 22}{\large a\tiny 31}{\,\scriptsize-\,}{\large a\tiny 11}{\large a\tiny 23}{\large a\tiny 32}{\,\scriptsize+\,}{\large a\tiny 12}{\large a\tiny 23}{\large a\tiny 31} \\ {\,\scriptsize =\,}{\enclose{roundedbox}{{\large a\tiny 11}{\large a\tiny 22}{\large a\tiny 33}{\,\scriptsize+\,}{\large a\tiny 12}{\large a\tiny 23}{\large a\tiny 31}{\,\scriptsize+\,}{\large a\tiny 13}{\large a\tiny 21}{\large a\tiny 32}{\,\scriptsize -\,}{\large a\tiny 13}{\large a\tiny 22}{\large a\tiny 31}{\,\scriptsize-\,}{\large a\tiny 12}{\large a\tiny 21}{\large a\tiny 33}{\,\scriptsize-\,}{\large a\tiny 11}{\large a\tiny 23}{\large a\tiny 32}}}\end{array}\right. \\[30pt] \overset{\:\:\small n}{\underset{\tiny I\:=\:1}{\LARGE\varSigma}}{\large a\tiny I\,j}\:{\large \tilde{a}\tiny I\,j}\hspace{5pt}\left\{\begin{array}{l}\Rightarrow\small\textsf{第$2$列の成分$\large a\tiny I\,{\color{red}2}$}\left\{\begin{array}{l}{\large a\tiny1{\color{red}2}} \\ {\large a\tiny2{\color{red}2}} \hspace{10pt}\begin{vmatrix}{a\tiny 11}&{\color{red}a\tiny 12}&{a\tiny 13} \\ {a\tiny 21}&\color{red}{a\tiny 22}&{a\tiny 23} \\ {a\tiny 31}&\color{red}{a\tiny 32}&{a\tiny 33}\end{vmatrix} \\ {\large a\tiny3{\color{red}2}}\end{array}\right.\hspace{10pt} \small\textsf{第$2$列の余因子}\left\{\begin{array}{l}{\large\tilde a\tiny1{\color{red} 2}}{\,\scriptsize =\,}({\scriptsize -}{\small 1})^{\tiny 1\,+\,\color{red}2 }\,\times{M\tiny1{\color{red} 2}}{\,\scriptsize =\,}{{\scriptsize -}\small 1}\times \begin{vmatrix} a\tiny 21& a\tiny 23 \\ a\tiny 31& a\tiny 33 \end{vmatrix}{\,\scriptsize =\,}{\scriptsize -\,}\large a\tiny 21\large a\tiny 33{\,\scriptsize +\,}\large a\tiny 23\large a\tiny 31 \\ {\large\tilde a\tiny2{\color{red} 2}}{\,\scriptsize =\,}({\scriptsize -}{\small 1})^{\tiny 2\,+\,\color{red} 2}\,\times{M\tiny 2{\color{red} 2}}{\,\scriptsize =\,}\hspace{6pt}{\small 1}\times \begin{vmatrix} a\tiny 11& a\tiny 13 \\ a\tiny 31& a\tiny 33 \end{vmatrix}{\,\scriptsize =\,}\hspace{8pt}\large a\tiny 11\large a\tiny 33{\,\scriptsize -\,}\large a\tiny 13\large a\tiny 31 \\ {\large\tilde a\tiny3{\color{red} 2}}{\,\scriptsize =\,}({\scriptsize -}{\small 1})^{\tiny 3\,+\,\color{red}2}\,\times{M\tiny3{\color{red} 2}}{\,\scriptsize =\,}{{\scriptsize -}\small 1}\times \begin{vmatrix} a\tiny 11& a\tiny 13 \\ a\tiny 21& a\tiny 23 \end{vmatrix}{\,\scriptsize =\,}{\scriptsize -\,}\large a\tiny 11\large a\tiny 23{\,\scriptsize +\,}\large a\tiny 13\large a\tiny 21 \end{array}\right. \\[10pt] {\,\scriptsize=\,}{\large a\tiny 12}({\scriptsize-\,}{\large a\tiny 21}{\large a\tiny 33}{\,\scriptsize+\,}{\large a\tiny 23}{\large a\tiny 31}){\,\scriptsize+\,}{\large a\tiny 22}({\large a\tiny 11}{\large a\tiny 33}{\,\scriptsize-\,}{\large a\tiny 13}{\large a\tiny 31}){\,\scriptsize+\,}{\large a\tiny 32}({\scriptsize-\,}{\large a\tiny 11}{\large a\tiny 23}{\,\scriptsize+\,}{\large a\tiny 13}{\large a\tiny 21}) \\{\,\scriptsize=\,}{\scriptsize-\,}{\large a\tiny 12}{\large a\tiny 21}{\large a\tiny 33}{\,\scriptsize+\,}{\large a\tiny 12}{\large a\tiny 23}{\large a\tiny 31}{\,\scriptsize+\,}{\large a\tiny 11}{\large a\tiny 22}{\large a\tiny 33}{\,\scriptsize-\,}{\large a\tiny 13}{\large a\tiny 22}{\large a\tiny 31}{\,\scriptsize-\,}{\large a\tiny 11}{\large a\tiny 23}{\large a\tiny 32}{\,\scriptsize+\,}{\large a\tiny 13}{\large a\tiny 21}{\large a\tiny 32} \\ {\,\scriptsize =\,}{\enclose{roundedbox}{{\large a\tiny 11}{\large a\tiny 22}{\large a\tiny 33}{\,\scriptsize+\,}{\large a\tiny 12}{\large a\tiny 23}{\large a\tiny 31}{\,\scriptsize+\,}{\large a\tiny 13}{\large a\tiny 21}{\large a\tiny 32}{\,\scriptsize -\,}{\large a\tiny 13}{\large a\tiny 22}{\large a\tiny 31}{\,\scriptsize-\,}{\large a\tiny 12}{\large a\tiny 21}{\large a\tiny 33}{\,\scriptsize-\,}{\large a\tiny 11}{\large a\tiny 23}{\large a\tiny 32}}} \end{array}\right.\end{cases}}$$
\(4\times4\) 行列式の余因子展開
$$det(A)=\begin{vmatrix}{\large a}{\tiny 11}&{\large a}{\tiny 12}&{\large a}{\tiny 13}&{\large a}{\tiny 14} \\ {\large a}{\tiny 21}&{\large a}{\tiny 22}&{\large a}{\tiny 23}&{\large a}{\tiny 24} \\ {\large a}{\tiny 31}&{\large a}{\tiny 32}&{\large a}{\tiny 33}&{\large a}{\tiny 34} \\ {\large a}{\tiny 41}&{\large a}{\tiny 42}&{\large a}{\tiny 43}&{\large a}{\tiny 44}\end{vmatrix}$$
\(\begin{array}{l}\textsf{余因子$\large\tilde a\tiny{\color{red}1}\,J$(どの行(または列)をとって展開しても行列式の値は同じである。)の第$1$行目で展開した解法を記する。} \\[5pt] \hspace{10pt}\textsf{なお、列の余因子展開は割愛させて頂く。}\end{array}\)
$$det(A)=\overset{\:\:\small n}
{\underset{\tiny J\:=\:1}{\LARGE\varSigma}}{\large a\tiny {\color{red}1}\,J}\:{\large \tilde{a}\tiny {\color{red}1}\,J}\hspace{5pt}({\scriptsize =}\:{\large a\tiny {\color{red}1}1}\,{\large \tilde{a}\tiny{\color{red}1}1}{\:\scriptsize +\:}{\large a\tiny {\color{red}1}2}\,{\large \tilde{a}\tiny{\color{red}1}2}
{\:\scriptsize +\:}{\large a\tiny {\color{red}1}3}\,{\large \tilde{a}\tiny{\color{red}1}3}{\:\scriptsize +\:}{\large a\tiny {\color{red}1}4}\,{\large \tilde{a}\tiny{\color{red}1}4})$$
$$det(A)={\large a\tiny{\color{red}1}1}\times({\small -1})^{\tiny{\color{red}1}\,+\,1}\times\begin{vmatrix}{a}{\tiny 22}&{a}{\tiny 23}&{a}{\tiny 24} \\ {a}{\tiny 32}&{a}{\tiny 33}&{a}{\tiny 34} \\ {a}{\tiny 42}&{a}{\tiny 43}&{a}{\tiny 44}\end{vmatrix}{\,\small +\,}{\large a\tiny{\color{red}1}2}\times({\small -1})^{\tiny{\color{red}1}\,+\,2}\times\begin{vmatrix}{a}{\tiny 21}&{a}{\tiny 23}&{a}{\tiny 24} \\ {a}{\tiny 31}&{a}{\tiny 33}&{a}{\tiny 34} \\ {a}{\tiny 41}&{a}{\tiny 43}&{a}{\tiny 44}\end{vmatrix}{\,\small +\,}{\large a\tiny{\color{red}1}3}\times({\small -1})^{\tiny{\color{red}1}\,+\,3}\times\begin{vmatrix}{a}{\tiny 21}&{a}{\tiny 22}&{a}{\tiny 24} \\ {a}{\tiny 31}&{a}{\tiny 32}&{a}{\tiny 34} \\ {a}{\tiny 41}&{a}{\tiny 42}&{a}{\tiny 44}\end{vmatrix}{\,\small +\,}{\large a\tiny{\color{red}1}4}\times({\small -1})^{\tiny{\color{red}1}\,+\,4}\times\begin{vmatrix}{a}{\tiny 21}&{a}{\tiny 22}&{a}{\tiny 23} \\ {a}{\tiny 31}&{a}{\tiny 32}&{a}{\tiny 33} \\ {a}{\tiny 41}&{a}{\tiny 42}&{a}{\tiny 43}\end{vmatrix}$$
\(\begin{array}{l}{\class{Boldfont}{\large =}}\hspace{5pt}{a\tiny{\color{red}1}1}{\small\times}({\small -1})^{\tiny{\color{red}1}\,+\,1}{\small\,\times}\left[a{\tiny 22}{\small\times}{\tilde{a}\tiny 22}\left\{\!{\scriptsize =\,}({\scriptsize -}{\small 1})^{\tiny 2\,+\,2}\,{\,\small\times}{\small M\tiny 22}{\scriptsize\, =\,}{\small 1}{\small\times\, }\begin{vmatrix} a\tiny 33\hspace{-5pt}&\hspace{-5pt} a\tiny 34 \\ a\tiny 43\hspace{-5pt}&\hspace{-5pt} a\tiny 44 \end{vmatrix}{\scriptsize \,=\,}{a\tiny 33}{a\tiny 44}{\scriptsize -}{a\tiny 34}{a\tiny 43} \right\} \right.\\ \hspace{100pt}\left.{\small \,+\,}a{\tiny 23}{\small\times}{\tilde{a}\tiny 23}\left\{\!{\scriptsize =\,}({\scriptsize -}{\small 1})^{\tiny 2\,+\,3}\,{\small\,\times}{\small M\tiny 23}{\scriptsize\, =\,}{\small -1}{\small\times}\begin{vmatrix} a\tiny 32\hspace{-5pt}&\hspace{-5pt} a\tiny 34 \\ a\tiny 42\hspace{-5pt}&\hspace{-5pt} a\tiny 44 \end{vmatrix}{\scriptsize\, =\,}{\scriptsize -}({a\tiny 32}{a\tiny 44}{\scriptsize -}{a\tiny 34}{a\tiny 42}) \right\}{\small \,+\,}a{\tiny 24}{\small\times}{\tilde{a}\tiny 24}\left\{\!{\scriptsize =\,}({\scriptsize -}{\small 1})^{\tiny 2\,+\,4}\,{\small\,\times}{\small M\tiny 24}{\scriptsize\, =\,}{\small 1}{\small\times}\begin{vmatrix} a\tiny 32\hspace{-5pt}&\hspace{-5pt} a\tiny 33 \\ a\tiny 42\hspace{-5pt}&\hspace{-5pt} a\tiny 43 \end{vmatrix}{\scriptsize \,=\,}{a\tiny 32}{a\tiny 43}{\scriptsize -}{a\tiny 33}{a\tiny 42} \right\}\right]\end{array}\)
\(\begin{array}{l}{\small+\,}{a\tiny{\color{red}1}2}{\small\times}({\small -1})^{\tiny{\color{red}1}\,+\,2}{\,\small\times}\left[a{\tiny 21}{\small\times}{\tilde{a}\tiny 21}\left\{\!{\scriptsize =\,}({\scriptsize -}{\small 1})^{\tiny 2\,+\,1}\,{\,\small\times}{\small M\tiny 21}{\scriptsize\, =\,}{\small -1}{\small\times}\begin{vmatrix} a\tiny 33\hspace{-5pt}&\hspace{-5pt} a\tiny 34 \\ a\tiny 43\hspace{-5pt}&\hspace{-5pt} a\tiny 44 \end{vmatrix}{\scriptsize \,=\,}{\scriptsize -}({a\tiny 33}{a\tiny 44}{\scriptsize -}{a\tiny 34}{a\tiny 43}) \right\}\right. \\ \hspace{100pt}\left.{\small \,+\,}a{\tiny 23}{\small\times}{\tilde{a}\tiny 23}\left\{\!{\scriptsize =\,}({\scriptsize -}{\small 1})^{\tiny 2\,+\,3}\,{\small\,\times}{\small M\tiny 23}{\scriptsize\, =\,}{\small -1}{\small\times}\begin{vmatrix} a\tiny 31\hspace{-5pt}&\hspace{-5pt} a\tiny 34 \\ a\tiny 41\hspace{-5pt}&\hspace{-5pt} a\tiny 44 \end{vmatrix}{\scriptsize \,=\,}{\scriptsize -}({a\tiny 31}{a\tiny 44}{\scriptsize -}{a\tiny 34}{a\tiny 41}) \right\}{\small \,+\,}a{\tiny 24}{\small\times}{\tilde{a}\tiny 24}\left\{\!{\scriptsize =\!}({\scriptsize -}{\small 1})^{\tiny 2\,+\,4}\,{\small\,\times}{\small M\tiny 24}{\scriptsize\, =\,}{\small 1}{\small\times}\begin{vmatrix} a\tiny 31\hspace{-5pt}&\hspace{-5pt} a\tiny 33 \\ a\tiny 41\hspace{-5pt}&\hspace{-5pt} a\tiny 43 \end{vmatrix}{\scriptsize \,=\,}{a\tiny 31}{a\tiny 43}{\scriptsize -}{a\tiny 33}{a\tiny 41} \right\}\right]\end{array}\)
\(\begin{array}{l}{\small+\,}{a\tiny{\color{red}1}3}{\small\times}({\small -1})^{\tiny{\color{red}1}\,+\,3}{\,\small\times}\left[a{\tiny 21}{\small\times}{\tilde{a}\tiny 21}\left\{\!{\scriptsize =\,}({\scriptsize -}{\small 1})^{\tiny 2\,+\,1}\,{\,\small\times}{\small M\tiny 21}{\scriptsize\, =\,}{\small -1}{\small\times}\begin{vmatrix} a\tiny 32\hspace{-5pt}&\hspace{-5pt} a\tiny 34 \\ a\tiny 42\hspace{-5pt}&\hspace{-5pt} a\tiny 44 \end{vmatrix}{\scriptsize \,=\,}{\scriptsize -}({a\tiny 32}{a\tiny 44}{\scriptsize -}{a\tiny 34}{a\tiny 42}) \right\}\right. \\ \hspace{100pt}\left.{\small \,+\,}a{\tiny 22}{\small\times}{\tilde{a}\tiny 22}\left\{\!{\scriptsize =\,}({\scriptsize -}{\small 1})^{\tiny 2\,+\,2}\,{\small\,\times}{\small M\tiny 22}{\scriptsize\, =\,}{\small 1}{\small\times}\begin{vmatrix} a\tiny 31\hspace{-5pt}&\hspace{-5pt} a\tiny 34 \\ a\tiny 41\hspace{-5pt}&\hspace{-5pt} a\tiny 44 \end{vmatrix}{\scriptsize \,=\,}{a\tiny 31}{a\tiny 44}{\scriptsize -}{a\tiny 34}{a\tiny 41} \right\}{\small \,+\,}a{\tiny 24}{\small\times}{\tilde{a}\tiny 24}\left\{\!{\scriptsize =\,}({\scriptsize -}{\small 1})^{\tiny 2\,+\,4}\,{\small\,\times}{\small M\tiny 24}{\scriptsize\, =\,}{\small 1}{\small\times}\begin{vmatrix} a\tiny 31\hspace{-5pt}&\hspace{-5pt} a\tiny 32 \\ a\tiny 41\hspace{-5pt}&\hspace{-5pt} a\tiny 42 \end{vmatrix}{\scriptsize \,=\,}{a\tiny 31}{a\tiny 42}{\scriptsize -}{a\tiny 32}{a\tiny 41} \right\}\right]\end{array}\)
\(\begin{array}{l}{\small+\,}{a\tiny{\color{red}1}4}{\small\times}({\small -1})^{\tiny{\color{red}1}\,+\,4}{\,\small\times}\left[a{\tiny 21}{\small\times}{\tilde{a}\tiny 21}\left\{\!{\scriptsize =\,}({\scriptsize -}{\small 1})^{\tiny 2\,+\,1}\,{\,\small\times}{\small M\tiny 21}{\scriptsize\, =\,}{\small -1}{\small\times}\begin{vmatrix} a\tiny 32\hspace{-5pt}&\hspace{-5pt} a\tiny 33 \\ a\tiny 42\hspace{-5pt}&\hspace{-5pt} a\tiny 43 \end{vmatrix}{\scriptsize \,=\,}{\scriptsize -}({a\tiny 32}{a\tiny 43}{\scriptsize -}{a\tiny 33}{a\tiny 42}) \right\}\right. \\ \hspace{100pt}\left.{\small \,+\,}a{\tiny 22}{\small\times}{\tilde{a}\tiny 22}\left\{\!{\scriptsize =\,}({\scriptsize -}{\small 1})^{\tiny 2\,+\,2}\,{\small\,\times}{\small M\tiny 22}{\scriptsize\, =\,}{\small 1}{\small\times}\begin{vmatrix} a\tiny 31\hspace{-5pt}&\hspace{-5pt} a\tiny 33 \\ a\tiny 41\hspace{-5pt}&\hspace{-5pt} a\tiny 43 \end{vmatrix}{\scriptsize \,=\,}{a\tiny 31}{a\tiny 43}{\scriptsize -}{a\tiny 33}{a\tiny 41} \right\}{\small \,+\,}a{\tiny 23}{\small\times}{\tilde{a}\tiny 23}\left\{\!{\scriptsize =\,}({\scriptsize -}{\small 1})^{\tiny 2\,+\,3}\,{\small\,\times}{\small M\tiny 23}{\scriptsize\, =\,}{\small -1}{\small\times}\begin{vmatrix} a\tiny 31\hspace{-5pt}&\hspace{-5pt} a\tiny 32 \\ a\tiny 41\hspace{-5pt}&\hspace{-5pt} a\tiny 42 \end{vmatrix}{\scriptsize \,=\,}{\scriptsize -}({a\tiny 31}{a\tiny 42}{\scriptsize -}{a\tiny 32}{a\tiny 41}) \right\}\right]\end{array}\)
\(\begin{array}{l}\,\,\hspace{-10pt}{\class{Boldfont}{\large =}}\hspace{5pt}{a\tiny 11}\left\{\hspace{7pt}{a\tiny 22}({a\tiny 33}{a\tiny 44}{\scriptsize -\,}{a\tiny 34}{a\tiny 33}){\,\scriptsize -\,}{a\tiny 23}({a\tiny 32}{a\tiny 44}{\scriptsize -\,}{a\tiny 34}{a\tiny 42}){\,\scriptsize +\,}{a\tiny 22}({a\tiny 33}{a\tiny 44}{\scriptsize -\,}{a\tiny 34}{a\tiny 33})\right\} \\ {\,\scriptsize -\,}{a\tiny 12}\left\{{\scriptsize -\,}{a\tiny 21}({a\tiny 33}{a\tiny 44}{\scriptsize -\,}{a\tiny 34}{a\tiny 43}){\,\scriptsize -\,}{a\tiny 23}({a\tiny 31}{a\tiny 44}{\scriptsize -\,}{a\tiny 34}{a\tiny 41}){\,\scriptsize +\,}{a\tiny 24}({a\tiny 31}{a\tiny 43}{\scriptsize -\,}{a\tiny 33}{a\tiny 41})\right\} \\ {\,\scriptsize +\,}{a\tiny 13}\left\{{\scriptsize -\,}{a\tiny 21}({a\tiny 32}{a\tiny 44}{\scriptsize -\,}{a\tiny 34}{a\tiny 42}){\,\scriptsize +\,}{a\tiny 22}({a\tiny 31}{a\tiny 44}{\scriptsize -\,}{a\tiny 34}{a\tiny 41}){\,\scriptsize +\,}{a\tiny 24}({a\tiny 31}{a\tiny 42}{\scriptsize -\,}{a\tiny 32}{a\tiny 41})\right\} \\ {\,\scriptsize -\,}{a\tiny 14}\left\{{\scriptsize -\,}{a\tiny 21}({a\tiny 32}{a\tiny 43}{\scriptsize -\,}{a\tiny 33}{a\tiny 42}){\,\scriptsize +\,}{a\tiny 22}({a\tiny 31}{a\tiny 43}{\scriptsize -\,}{a\tiny 33}{a\tiny 41}){\,\scriptsize -\,}{a\tiny 23}({a\tiny 31}{a\tiny 42} {\scriptsize -\,}{a\tiny 32}{a\tiny 41})\right\}\end{array}\)
\(\hspace{30pt}{\class{Boldfont}{\large =}}\hspace{30pt}{\enclose{roundedbox}{\begin{array}{l}\hspace{7pt}
a{\tiny{11}}\,a{\tiny{22}}\,a{\tiny{33}}\,a{\tiny{44}}\:{\scriptsize +}\,a{\tiny{11}}\,a{\tiny{23}}\,a{\tiny{34}}\,a{\tiny{42}}\:{\scriptsize +}\, a{\tiny{11}}\,a{\tiny{24}}\,a{\tiny{32}}\,a{\tiny{43}}\:{\scriptsize -}\:a{\tiny{11}}\,a{\tiny{22}}\,a{\tiny{34}}\,a{\tiny
{43}}\:{\scriptsize -}\: a{\tiny{11}}\,a{\tiny{23}}\,a{\tiny{32}}\,a{\tiny{44}}\:{\scriptsize -}\:a{\tiny{11}}\,a{\tiny{24}}\,a{\tiny{33}}\,a{\tiny{42}} \\
{\scriptsize -}\, a{\tiny{12}}\,a{\tiny{21}}\,a{\tiny{33}}\,a{\tiny{44}}\:{\scriptsize -}\:a{\tiny{12}}\,a{\tiny{23}}\,a{\tiny{34}}\,a{\tiny{41}}\:{\scriptsize -}\, a{\tiny{12}}
\,a{\tiny{24}}\,a{\tiny{31}}\,a{\tiny{43}}\:{\scriptsize +}\:a{\tiny{12}}\,a{\tiny{21}}\,a{\tiny
{34}}\,a{\tiny{43}}\:{\scriptsize +}\, a{\tiny{12}}\,a{\tiny{23}}\,a{\tiny{31}}\,a{\tiny{44}}\:
{\scriptsize +}\:a{\tiny{12}}\,a{\tiny{24}}\,a{\tiny{33}}\,a{\tiny{41}} \\{\scriptsize +}\, a{\tiny{13}}\,a{\tiny{21}}\,a{\tiny{32}}\,a{\tiny{44}}\:{\scriptsize +}\:a{\tiny
{13}}\,a{\tiny{22}}\,a{\tiny{34}}\,a{\tiny{41}}\:{\scriptsize +}\, a{\tiny{13}}\,a{\tiny{24}}\,
a{\tiny{31}}\,a{\tiny{42}}\:{\scriptsize -}\:a{\tiny{13}}\,a{\tiny{21}}\,a{\tiny{34}}\,a{\tiny{42}}\:{\scriptsize -}\, a{\tiny{13}}\,a{\tiny{22}}\,a{\tiny{31}}\,a{\tiny{44}}\:{\scriptsize -}\:
a{\tiny{13}}\,a{\tiny{24}}\,a{\tiny{32}}\,a{\tiny{41}} \\
{\scriptsize -}\: a{\tiny{14}}\,a{\tiny{21}}\,a{\tiny{32}}\,a{\tiny{43}}\:{\scriptsize -}\: a{\tiny{14}}\,a{\tiny{22}}\,a{\tiny{33}}\,a{\tiny{41}}\:{\scriptsize -}\, a{\tiny{14}}\,a{\tiny{23}}\,a{\tiny{31}}\,a{\tiny{42}}\:{\scriptsize +}\, a{\tiny{14}}\,a{\tiny{21}}\,
a{\tiny{33}}\,a{\tiny{42}}\:{\scriptsize +}\: a{\tiny{14}}\,a{\tiny{22}}\,a{\tiny{31}}\,
a{\tiny{43}}\:{\scriptsize +}\, a{\tiny{14}}\,a{\tiny{23}}\,a{\tiny{32}}\,a{\tiny{41}}\end{array}}\hspace{-515pt}}\)
\(5\)次以上の正方行列の行列式も同様の手順で余因子展開し、解を求めていく。