余因子

$3$\displaystyle{n}$ 次の正方行列 $3$\displaystyle{A}$

$3$\displaystyle{\displaystyle{A=\left({\begin{array}\hspace{1}{a}_{11}&\cdots &\cdots &\hspace{1}{a}_{ 1n }& \vspace{6}\\ \vdots &\ddots & &\vdots & \vspace{6}\\ \vdots & &\ddots &\vdots & \vspace{6}\\ \hspace{1}{a}_{ n1 }&\cdots &\cdots &\hspace{1}{a}_{ nn }& \vspace{6}\\ \end{array}}\right)}}$

の $3$\displaystyle{\hspace{1}{a}_{ ij }}$ の余因子( $3$\displaystyle{\hspace{1}{\tilde{a}}_{ ij }}$ と書く)を

$3$\displaystyle{\hspace{1}{\tilde{a}}_{ ij }={ $ - 1 $ }^{ i+ j }\left({\begin{array}\hspace{1}{a}_{11}&\cdots &\hspace{1}{a}_{ 1$j- 1$ }&\hspace{1}{a}_{ 1$j+ 1$ }&\cdots &\hspace{1}{a}_{ 1n }& \vspace{6}\\ \vdots & &\vdots &\vdots & & & \vspace{6}\\ \hspace{1}{a}_{ $i- 1$1 }&\cdots & & &\cdots &\hspace{1}{a}_{ $i- 1$1 }& \vspace{6}\\ \hspace{1}{a}_{ $i+ 1$1 }&\cdots & & &\cdots &\hspace{1}{a}_{ $i+ 1$1 }& \vspace{6}\\ \vdots & &\vdots &\vdots & & & \vspace{6}\\ \hspace{1}{a}_{ n1 }&\cdots &\hspace{1}{a}_{ n$j- 1$ }&\hspace{1}{a}_{ n$j+ 1$ }&\cdots &\hspace{1}{a}_{ nn }& \vspace{6}\\ \end{array}}\right)}$

と定める．つまり， $3$\displaystyle{A}$ の行列式 $3$\displaystyle{ \|A\| }$ から $3$\displaystyle{j}$ 列の $3$\displaystyle{\hspace{1}{a}_{ 1j }\hspace{5}\hspace{10} }$ ～ $3$\displaystyle{\hspace{1}{a}_{ nj }\hspace{5}\hspace{10} }$ までの成分と， $3$\displaystyle{i}$ 行の $3$\displaystyle{\hspace{1}{a}_{ i1 }}$ ～ $3$\displaystyle{\hspace{1}{a}_{ in }}$ 　までの成分を削除し得られる $3$\displaystyle{ $ n- 1 $ }$ 次の行列式に

$3$\displaystyle{{ $ - 1 $ }^{ i+j }}$ 　をかけたものを余因子という．

■具体例

$3$\displaystyle{ A=\left({\begin{array}1&2&3& \vspace{6}\\ 4&5&6& \vspace{6}\\ 7&8&9& \vspace{6}\\ \end{array}}\right) }$

$3$\displaystyle{ \hspace{1}{\tilde{a}}_{11} }$ $3$\displaystyle{ ={\left({ - 1 }\right)}^{ 1+1 }\left({\begin{array}5&6& \vspace{6}\\ 8&9& \vspace{6}\\ \end{array}}\right) }$

$3$\displaystyle{ =\left({ 5\times 9- 8\times 6 }\right) }$

$3$\displaystyle{ =- 3 }$

$3$\displaystyle{ \hspace{1}{\tilde{a}}_{12} }$ $3$\displaystyle{ ={\left({ - 1 }\right)}^{ 1+2 }\left({\begin{array}4&6& \vspace{6}\\ 7&9& \vspace{6}\\ \end{array}}\right) }$

$3$\displaystyle{ =\left({ 4\times 9- 7\times 6 }\right) }$

$3$\displaystyle{ =- 6 }$

$3$\displaystyle{ \hspace{1}{\tilde{a}}_{13} }$ $3$\displaystyle{ ={\left({ - 1 }\right)}^{ 1+3 }\left({\begin{array}4&5& \vspace{6}\\ 7&8& \vspace{6}\\ \end{array}}\right) }$

$3$\displaystyle{ =\left({ 4\times 8- 7\times 5 }\right) }$

$3$\displaystyle{ =- 3 }$

$3$\displaystyle{ \hspace{1}{\tilde{a}}_{21} }$ $3$\displaystyle{ ={\left({ - 1 }\right)}^{ 2+1 }\left({\begin{array}2&3& \vspace{6}\\ 8&9& \vspace{6}\\ \end{array}}\right) }$

$3$\displaystyle{ =\left({ 2\times 9- 8\times 3 }\right) }$

$3$\displaystyle{ =- 6 }$

$3$\displaystyle{ \hspace{1}{\tilde{a}}_{22} }$ $3$\displaystyle{ ={\left({ - 1 }\right)}^{ 2+2 }\left({\begin{array}1&3& \vspace{6}\\ 7&9& \vspace{6}\\ \end{array}}\right) }$

$3$ =\left({ 1\times 9- 7\times 3 }\right)$

$3$\displaystyle{ =- 12 }$

$3$\displaystyle{ \hspace{1}{\tilde{a}}_{23} }$ $3$\displaystyle{ ={\left({ - 1 }\right)}^{ 2+3 }\left({\begin{array}1&2& \vspace{6}\\ 7&8& \vspace{6}\\ \end{array}}\right) }$

$3$\displaystyle{ =\left({ 1\times 8- 7\times 2 }\right) }$

$3$\displaystyle{ =- 6 }$

$3$\displaystyle{ \hspace{1}{\tilde{a}}_{31} }$ $3$\displaystyle{ ={\left({ - 1 }\right)}^{ 3+1 }\left({\begin{array}2&3& \vspace{6}\\ 5&6& \vspace{6}\\ \end{array}}\right) }$

$3$\displaystyle{ =\left({ 2\times 6- 5\times 3 }\right) }$

$3$\displaystyle{ =- 3 }$

$3$\displaystyle{ \hspace{1}{\tilde{a}}_{32} }$ $3$\displaystyle{ ={\left({ - 1 }\right)}^{ 3+2 }\left({\begin{array}1&3& \vspace{6}\\ 4&6& \vspace{6}\\ \end{array}}\right) }$

$3$\displaystyle{ =\left({ 1\times 6- 4\times 3 }\right) }$

$3$\displaystyle{ =- 6 }$

$3$\displaystyle{ \hspace{1}{\tilde{a}}_{33} }$ $3$\displaystyle{ ={\left({ - 1 }\right)}^{ 3+3 }\left({\begin{array}1&2& \vspace{6}\\ 4&5& \vspace{6}\\ \end{array}}\right) }$

$3$\displaystyle{ =\left({ 1\times 5- 4\times 2 }\right) }$

$3$\displaystyle{ =- 3 }$

ホーム>>カテゴリー分類>>行列>>線形代数>>余因子

学生スタッフ作成
最終更新日： 2023年2月9日

$3$\displaystyle{ \hspace{1}{\tilde{a}}_{11} }$	$3$\displaystyle{ ={\left({ - 1 }\right)}^{ 1+1 }\left({\begin{array}5&6& \vspace{6}\\ 8&9& \vspace{6}\\ \end{array}}\right) }$
	$3$\displaystyle{ =\left({ 5\times 9- 8\times 6 }\right) }$
	$3$\displaystyle{ =- 3 }$

$3$\displaystyle{ \hspace{1}{\tilde{a}}_{12} }$	$3$\displaystyle{ ={\left({ - 1 }\right)}^{ 1+2 }\left({\begin{array}4&6& \vspace{6}\\ 7&9& \vspace{6}\\ \end{array}}\right) }$
	$3$\displaystyle{ =\left({ 4\times 9- 7\times 6 }\right) }$
	$3$\displaystyle{ =- 6 }$

$3$\displaystyle{ \hspace{1}{\tilde{a}}_{13} }$	$3$\displaystyle{ ={\left({ - 1 }\right)}^{ 1+3 }\left({\begin{array}4&5& \vspace{6}\\ 7&8& \vspace{6}\\ \end{array}}\right) }$
	$3$\displaystyle{ =\left({ 4\times 8- 7\times 5 }\right) }$
	$3$\displaystyle{ =- 3 }$

$3$\displaystyle{ \hspace{1}{\tilde{a}}_{21} }$	$3$\displaystyle{ ={\left({ - 1 }\right)}^{ 2+1 }\left({\begin{array}2&3& \vspace{6}\\ 8&9& \vspace{6}\\ \end{array}}\right) }$
	$3$\displaystyle{ =\left({ 2\times 9- 8\times 3 }\right) }$
	$3$\displaystyle{ =- 6 }$

$3$\displaystyle{ \hspace{1}{\tilde{a}}_{22} }$	$3$\displaystyle{ ={\left({ - 1 }\right)}^{ 2+2 }\left({\begin{array}1&3& \vspace{6}\\ 7&9& \vspace{6}\\ \end{array}}\right) }$
	$3$ =\left({ 1\times 9- 7\times 3 }\right)$
	$3$\displaystyle{ =- 12 }$

$3$\displaystyle{ \hspace{1}{\tilde{a}}_{23} }$	$3$\displaystyle{ ={\left({ - 1 }\right)}^{ 2+3 }\left({\begin{array}1&2& \vspace{6}\\ 7&8& \vspace{6}\\ \end{array}}\right) }$
	$3$\displaystyle{ =\left({ 1\times 8- 7\times 2 }\right) }$
	$3$\displaystyle{ =- 6 }$

$3$\displaystyle{ \hspace{1}{\tilde{a}}_{31} }$	$3$\displaystyle{ ={\left({ - 1 }\right)}^{ 3+1 }\left({\begin{array}2&3& \vspace{6}\\ 5&6& \vspace{6}\\ \end{array}}\right) }$
	$3$\displaystyle{ =\left({ 2\times 6- 5\times 3 }\right) }$
	$3$\displaystyle{ =- 3 }$

$3$\displaystyle{ \hspace{1}{\tilde{a}}_{32} }$	$3$\displaystyle{ ={\left({ - 1 }\right)}^{ 3+2 }\left({\begin{array}1&3& \vspace{6}\\ 4&6& \vspace{6}\\ \end{array}}\right) }$
	$3$\displaystyle{ =\left({ 1\times 6- 4\times 3 }\right) }$
	$3$\displaystyle{ =- 6 }$

$3$\displaystyle{ \hspace{1}{\tilde{a}}_{33} }$	$3$\displaystyle{ ={\left({ - 1 }\right)}^{ 3+3 }\left({\begin{array}1&2& \vspace{6}\\ 4&5& \vspace{6}\\ \end{array}}\right) }$
	$3$\displaystyle{ =\left({ 1\times 5- 4\times 2 }\right) }$
	$3$\displaystyle{ =- 3 }$