La regla de Cramer és un teorema , en àlgebra lineal, que dona la solució d'un sistema lineal d'equacions compatible determinat en termes de determinants . Rep aquest nom en honor de Gabriel Cramer , el qual va presentar aquest resultat en l'obra Introduction à l'analyse des lignes Courbes algebraiques (Introducció a l'anàlisi de les cobes algebraiques) publicat en 1750 ,[ 1] Colin Maclaurin ja l'havia publicat el 1748 (i possiblement ja el coneixia des del 1729)[ 2] [ 3]
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{\displaystyle A{\vec {x}}={\vec {b}}}
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{\displaystyle A={\begin{pmatrix}a_{11}&\cdots &a_{1n}\\\vdots &\ddots &\vdots \\a_{n1}&\cdots &a_{nn}\end{pmatrix}};{\vec {x}}={\begin{pmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{pmatrix}};{\vec {b}}={\begin{pmatrix}b_{1}\\b_{2}\\\vdots \\b_{n}\end{pmatrix}}}
Si
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{\displaystyle \det(A)\neq 0}
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{\displaystyle x_{j}={\det(A_{j}) \over \det(A)},\quad j=1,\dots ,n,}
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{\displaystyle A_{j}={\begin{pmatrix}a_{11}&\cdots &a_{1,j-1}&b_{1}&a_{1,j+1}&\cdots &a_{1n}\\a_{21}&\ddots &&&&&\vdots \\\vdots &&\ddots &&&&\vdots \\\vdots &&&\ddots &&&\vdots \\\vdots &&&&\ddots &&\vdots \\\vdots &&&&&\ddots &a_{n-1n}\\a_{n1}&\cdots &a_{n,j-1}&b_{n}&a_{n,j+1}&\cdots &a_{nn}\end{pmatrix}}}
Donat el següent sistema d'equacions de dimensió 2x2
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{\displaystyle {\begin{cases}ax+by={\color {red}e}\\cx+dy={\color {red}f}\end{cases}}}
La seva forma matricial és
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{\displaystyle {\begin{pmatrix}a&b\\c&d\end{pmatrix}}{\begin{pmatrix}x\\y\end{pmatrix}}={\begin{pmatrix}{\color {red}e}\\{\color {red}f}\end{pmatrix}}}
La seva solució és, per la regla de Cramer,
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{\displaystyle x={\frac {\begin{vmatrix}{\color {red}e}&b\\{\color {red}f}&d\end{vmatrix}}{\begin{vmatrix}a&b\\c&d\end{vmatrix}}}={{\color {red}e}d-b{\color {red}f} \over ad-bc}}
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{\displaystyle y={\frac {\begin{vmatrix}a&{\color {red}e}\\c&{\color {red}f}\end{vmatrix}}{\begin{vmatrix}a&b\\c&d\end{vmatrix}}}={a{\color {red}f}-{\color {red}e}c \over ad-bc}}
Donat el següent sistema d'equcions de dimensión 3x3:
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{\displaystyle {\begin{cases}ax+by+cz={\color {red}j}\\dx+ey+fz={\color {red}k}\\gx+hy+iz={\color {red}l}\end{cases}}}
La seva forma matricial és
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{\displaystyle {\begin{pmatrix}a&b&c\\d&e&f\\g&h&i\end{pmatrix}}{\begin{pmatrix}{x}\\{y}\\{z}\end{pmatrix}}={\begin{pmatrix}{\color {red}j}\\{\color {red}k}\\{\color {red}l}\end{pmatrix}}}
La seva solució és, per la regla de Cramer,
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{\displaystyle x={\frac {\begin{vmatrix}{\color {red}j}&b&c\\{\color {red}k}&e&f\\{\color {red}l}&h&i\end{vmatrix}}{\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}};\quad y={\frac {\begin{vmatrix}a&{\color {red}j}&c\\d&{\color {red}k}&f\\g&{\color {red}l}&i\end{vmatrix}}{\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}},\quad z={\frac {\begin{vmatrix}a&b&{\color {red}j}\\d&e&{\color {red}k}\\g&h&{\color {red}l}\end{vmatrix}}{\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}}}
