三階線型方程式系
出典: フリー百科事典『ウィキペディア（Wikipedia）』 (2021/07/17 21:27 UTC 版)
別な例として次の線型方程式系 { 82 x 1 + 45 x 2 + 9 x 3 = 1 27 x 1 + 16 x 2 + 3 x 3 = 1 9 x 1 + 5 x 2 + 1 x 3 = 0 {\displaystyle {\begin{cases}{\color {blue}82}\,x_{1}+{\color {blue}45}\,x_{2}+{\color {blue}9}\,x_{3}={\color {OliveGreen}1}\\{\color {blue}27}\,x_{1}+{\color {blue}16}\,x_{2}+{\color {blue}3}\,x_{3}={\color {OliveGreen}1}\\{\color {blue}9}\,x_{1}+{\color {blue}5}\,x_{2}+{\color {blue}1}\,x_{3}={\color {OliveGreen}0}\\\end{cases}}} をとる。拡大係数行列は ( A ∣ b ) = [ 82 45 9 1 27 16 3 1 9 5 1 0 ] {\displaystyle ({\color {blue}A}\mid {\color {OliveGreen}b})=\left[{\begin{array}{ccc|c}{\color {blue}82}&{\color {blue}45}&{\color {blue}9}&{\color {OliveGreen}1}\\{\color {blue}27}&{\color {blue}16}&{\color {blue}3}&{\color {OliveGreen}1}\\{\color {blue}9}&{\color {blue}5}&{\color {blue}1}&{\color {OliveGreen}0}\end{array}}\right]} である。解をクラメルの法則に従って求めれば、 x 1 = det ( A 1 ) det ( A ) = | 1 45 9 1 16 3 0 5 1 | | 82 45 9 27 16 3 9 5 1 | = 1 1 = 1 , x 2 = det ( A 2 ) det ( A ) = | 82 1 9 27 1 3 9 0 1 | | 82 45 9 27 16 3 9 5 1 | = 1 1 = 1 , x 3 = det ( A 3 ) det ( A ) = | 82 45 1 27 16 1 9 5 0 | | 82 45 9 27 16 3 9 5 1 | = − 14 1 = − 14 {\displaystyle {\begin{aligned}x_{1}&={\frac {\det(A_{1})}{\det(A)}}={\tfrac {\begin{vmatrix}\color {OliveGreen}{1}&\color {blue}{45}&\color {blue}{9}\\\color {OliveGreen}{1}&\color {blue}{16}&\color {blue}{3}\\\color {OliveGreen}{0}&\color {blue}{5}&\color {blue}{1}\end{vmatrix}}{\begin{vmatrix}\color {blue}{82}&\color {blue}{45}&\color {blue}{9}\\\color {blue}{27}&\color {blue}{16}&\color {blue}{3}\\\color {blue}{9}&\color {blue}{5}&\color {blue}{1}\end{vmatrix}}}={\frac {1}{1}}=1,\\[10pt]x_{2}&={\frac {\det(A_{2})}{\det(A)}}={\tfrac {\begin{vmatrix}\color {blue}{82}&\color {OliveGreen}{1}&\color {blue}{9}\\\color {blue}{27}&\color {OliveGreen}{1}&\color {blue}{3}\\\color {blue}{9}&\color {OliveGreen}{0}&\color {blue}{1}\end{vmatrix}}{\begin{vmatrix}\color {blue}{82}&\color {blue}{45}&\color {blue}{9}\\\color {blue}{27}&\color {blue}{16}&\color {blue}{3}\\\color {blue}{9}&\color {blue}{5}&\color {blue}{1}\end{vmatrix}}}={\frac {1}{1}}=1,\\[10pt]x_{3}&={\frac {\det(A_{3})}{\det(A)}}={\tfrac {\begin{vmatrix}\color {blue}{82}&\color {blue}{45}&\color {OliveGreen}{1}\\\color {blue}{27}&\color {blue}{16}&\color {OliveGreen}{1}\\\color {blue}{9}&\color {blue}{5}&\color {OliveGreen}{0}\end{vmatrix}}{\begin{vmatrix}\color {blue}{82}&\color {blue}{45}&\color {blue}{9}\\\color {blue}{27}&\color {blue}{16}&\color {blue}{3}\\\color {blue}{9}&\color {blue}{5}&\color {blue}{1}\end{vmatrix}}}={\frac {-14}{1}}=-14\end{aligned}}} となる。
※この「三階線型方程式系」の解説は、「クラメルの公式」の解説の一部です。
「三階線型方程式系」を含む「クラメルの公式」の記事については、「クラメルの公式」の概要を参照ください。
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