クラメルの公式
■2元1次連立方程式の場合
連立方程式
を行列を用いて，
%5cleft({%5cbegin{array}x&%20%5cvspace{6}%5c%5c%20y&%20%5cvspace{6}%5c%5c%20%20%5cend{array}}%5cright)=%5cleft({%5cbegin{array}%5chspace{1}{c}_{1}&%20%5cvspace{6}%5c%5c%20%5chspace{1}{c}_{2}&%20%5cvspace{6}%5c%5c%20%20%5cend{array}}%5cright)%20%20})
と表す．
%20%20})
とおく．
=%5cleft({%5cbegin{array}%5chspace{1}{a}_{1}&%5chspace{1}{b}_{1}&%20%5cvspace{6}%5c%5c%20%5chspace{1}{a}_{2}&%5chspace{1}{b}_{2}&%20%5cvspace{6}%5c%5c%20%20%5cend{array}}%5cright)%5cneq%200%20%20})
のとき，連立方程式の解は，
%5chspace{2}}%5cleft({%5cbegin{array}%5chspace{1}{c}_{1}&%5chspace{1}{b}_{1}&%20%5cvspace{6}%5c%5c%20%5chspace{1}{c}_{2}&%5chspace{1}{b}_{2}&%20%5cvspace{6}%5c%5c%20%20%5cend{array}}%5cright)=%5cfrac{%5chspace{2}%5cleft({%5cbegin{array}%5chspace{1}{c}_{1}&%5chspace{1}{b}_{1}&%20%5cvspace{6}%5c%5c%20%5chspace{1}{c}_{2}&%5chspace{1}{b}_{2}&%20%5cvspace{6}%5c%5c%20%20%5cend{array}}%5cright)%5chspace{2}}{%5chspace{2}%5cleft({%5cbegin{array}%5chspace{1}{a}_{1}&%5chspace{1}{b}_{1}&%20%5cvspace{6}%5c%5c%20%5chspace{1}{a}_{2}&%5chspace{1}{b}_{2}&%20%5cvspace{6}%5c%5c%20%20%5cend{array}}%5cright)%5chspace{2}}%20%20})
%5chspace{2}}%5cleft({%5cbegin{array}%5chspace{1}{a}_{1}&%5chspace{1}{c}_{1}&%20%5cvspace{6}%5c%5c%20%5chspace{1}{a}_{2}&%5chspace{1}{c}_{2}&%20%5cvspace{6}%5c%5c%20%20%5cend{array}}%5cright)=%5cfrac{%5chspace{2}%5cleft({%5cbegin{array}%5chspace{1}{a}_{1}&%5chspace{1}{c}_{1}&%20%5cvspace{6}%5c%5c%20%5chspace{1}{a}_{2}&%5chspace{1}{c}_{2}&%20%5cvspace{6}%5c%5c%20%20%5cend{array}}%5cright)%5chspace{2}}{%5chspace{2}%5cleft({%5cbegin{array}%5chspace{1}{a}_{1}&%5chspace{1}{b}_{1}&%20%5cvspace{6}%5c%5c%20%5chspace{1}{a}_{2}&%5chspace{1}{b}_{2}&%20%5cvspace{6}%5c%5c%20%20%5cend{array}}%5cright)%5chspace{2}}%20%20})
で与えられる．
これらの解を表す式をクラメルの公式という．
■3元1次連立方程式の場合
連立方程式
を行列を用いて
と表す．
%20%20})
とおく．
=%5cleft({%5cbegin{array}%5chspace{1}{a}_{1}&%5chspace{1}{b}_{1}&%5chspace{1}{c}_{1}&%20%5cvspace{6}%5c%5c%20%5chspace{1}{a}_{2}&%5chspace{1}{b}_{2}&%5chspace{1}{c}_{2}&%20%5cvspace{6}%5c%5c%20%5chspace{1}{a}_{3}&%5chspace{1}{b}_{3}&%5chspace{1}{c}_{3}&%20%5cvspace{6}%5c%5c%20%20%5cend{array}}%5cright)%5cneq%200%20%20})
のとき，連立方程式の解は，
%5chspace{2}}%5cleft({%5cbegin{array}%5chspace{1}{d}_{1}&%5chspace{1}{b}_{1}&%5chspace{1}{c}_{1}&%20%5cvspace{6}%5c%5c%20%5chspace{1}{d}_{2}&%5chspace{1}{b}_{2}&%5chspace{1}{c}_{2}&%20%5cvspace{6}%5c%5c%20%5chspace{1}{d}_{3}&%5chspace{1}{b}_{3}&%5chspace{1}{c}_{3}&%20%5cvspace{6}%5c%5c%20%20%5cend{array}}%5cright)=%5cfrac{%5chspace{2}%5cleft({%5cbegin{array}%5chspace{1}{d}_{1}&%5chspace{1}{b}_{1}&%5chspace{1}{c}_{1}&%20%5cvspace{6}%5c%5c%20%5chspace{1}{d}_{2}&%5chspace{1}{b}_{2}&%5chspace{1}{c}_{2}&%20%5cvspace{6}%5c%5c%20%5chspace{1}{d}_{3}&%5chspace{1}{b}_{3}&%5chspace{1}{c}_{3}&%20%5cvspace{6}%5c%5c%20%20%5cend{array}}%5cright)%5chspace{2}}{%5chspace{2}%5cleft({%5cbegin{array}%5chspace{1}{a}_{1}&%5chspace{1}{b}_{1}&%5chspace{1}{c}_{1}&%20%5cvspace{6}%5c%5c%20%5chspace{1}{a}_{2}&%5chspace{1}{b}_{2}&%5chspace{1}{c}_{2}&%20%5cvspace{6}%5c%5c%20%5chspace{1}{a}_{3}&%5chspace{1}{b}_{3}&%5chspace{1}{c}_{3}&%20%5cvspace{6}%5c%5c%20%20%5cend{array}}%5cright)%5chspace{2}}%20%20})
%5chspace{2}}%5cleft({%5cbegin{array}%5chspace{1}{a}_{1}&%5chspace{1}{d}_{1}&%5chspace{1}{c}_{1}&%20%5cvspace{6}%5c%5c%20%5chspace{1}{a}_{2}&%5chspace{1}{d}_{2}&%5chspace{1}{c}_{2}&%20%5cvspace{6}%5c%5c%20%5chspace{1}{a}_{3}&%5chspace{1}{d}_{3}&%5chspace{1}{c}_{3}&%20%5cvspace{6}%5c%5c%20%20%5cend{array}}%5cright)=%5cfrac{%5chspace{2}%5cleft({%5cbegin{array}%5chspace{1}{a}_{1}&%5chspace{1}{d}_{1}&%5chspace{1}{c}_{1}&%20%5cvspace{6}%5c%5c%20%5chspace{1}{a}_{2}&%5chspace{1}{d}_{2}&%5chspace{1}{c}_{2}&%20%5cvspace{6}%5c%5c%20%5chspace{1}{a}_{3}&%5chspace{1}{d}_{3}&%5chspace{1}{c}_{3}&%20%5cvspace{6}%5c%5c%20%20%5cend{array}}%5cright)%5chspace{2}}{%5chspace{2}%5cleft({%5cbegin{array}%5chspace{1}{a}_{1}&%5chspace{1}{b}_{1}&%5chspace{1}{c}_{1}&%20%5cvspace{6}%5c%5c%20%5chspace{1}{a}_{2}&%5chspace{1}{b}_{2}&%5chspace{1}{c}_{2}&%20%5cvspace{6}%5c%5c%20%5chspace{1}{a}_{3}&%5chspace{1}{b}_{3}&%5chspace{1}{c}_{3}&%20%5cvspace{6}%5c%5c%20%20%5cend{array}}%5cright)%5chspace{2}}%20%20})
%5chspace{2}}%5cleft({%5cbegin{array}%5chspace{1}{a}_{1}&%5chspace{1}{b}_{1}&%5chspace{1}{d}_{1}&%20%5cvspace{6}%5c%5c%20%5chspace{1}{a}_{2}&%5chspace{1}{b}_{2}&%5chspace{1}{d}_{2}&%20%5cvspace{6}%5c%5c%20%5chspace{1}{a}_{3}&%5chspace{1}{b}_{3}&%5chspace{1}{d}_{3}&%20%5cvspace{6}%5c%5c%20%20%5cend{array}}%5cright)=%5cfrac{%5chspace{2}%5cleft({%5cbegin{array}%5chspace{1}{a}_{1}&%5chspace{1}{b}_{1}&%5chspace{1}{d}_{1}&%20%5cvspace{6}%5c%5c%20%5chspace{1}{a}_{2}&%5chspace{1}{b}_{2}&%5chspace{1}{d}_{2}&%20%5cvspace{6}%5c%5c%20%5chspace{1}{a}_{3}&%5chspace{1}{b}_{3}&%5chspace{1}{d}_{3}&%20%5cvspace{6}%5c%5c%20%20%5cend{array}}%5cright)%5chspace{2}}{%5chspace{2}%5cleft({%5cbegin{array}%5chspace{1}{a}_{1}&%5chspace{1}{b}_{1}&%5chspace{1}{c}_{1}&%20%5cvspace{6}%5c%5c%20%5chspace{1}{a}_{2}&%5chspace{1}{b}_{2}&%5chspace{1}{c}_{2}&%20%5cvspace{6}%5c%5c%20%5chspace{1}{a}_{3}&%5chspace{1}{b}_{3}&%5chspace{1}{c}_{3}&%20%5cvspace{6}%5c%5c%20%20%5cend{array}}%5cright)%5chspace{2}}%20%20})
で与えられる．
■n元1次連立方程式の場合
を行列を用いて
と表す．
%20%20})
とおく．
のとき，連立方程式の解は，
で与えられる．⇒導出
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最終更新日：2022年9月9日