Difference between revisions of "Cramer rule"
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| + | If the determinant $ D $ | ||
| + | of a square system of linear equations | ||
| + | |||
| + | $$ | ||
| + | \begin{array}{c} | ||
| + | |||
| + | a _ {11} x _ {1} + \dots + a _ {1n} x _ {n} = b _ {1} , | ||
| + | \\ | ||
| + | |||
| + | {\dots \dots \dots \dots } | ||
| + | \\ | ||
| + | |||
| + | a _ {n1} x _ {1} + \dots + a _ {nn} x _ {n} = b _ {n} | ||
| + | \end{array} | ||
| + | |||
| + | $$ | ||
does not vanish, then the system has a unique solution. This solution is given by the formulas | does not vanish, then the system has a unique solution. This solution is given by the formulas | ||
| − | + | $$ \tag{* } | |
| + | x _ {k} = \ | ||
| − | + | \frac{D _ {k} }{D} | |
| + | ,\ \ | ||
| + | k = 1 \dots n. | ||
| + | $$ | ||
| + | |||
| + | Here $ D _ {k} $ | ||
| + | is the determinant obtained from $ D $ | ||
| + | when the $ k $- | ||
| + | th column is replaced by the column of the free terms $ b _ {1} \dots b _ {n} $. | ||
| + | Formulas (*) are known as Cramer's formulas. They have been found by G. Cramer (see [[#References|[1]]]). | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> G. Cramer, "Introduction à l'analyse des lignes courbes" , Geneva (1750) pp. 657</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.G. Kurosh, "Higher algebra" , MIR (1972) (Translated from Russian)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> G. Cramer, "Introduction à l'analyse des lignes courbes" , Geneva (1750) pp. 657</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.G. Kurosh, "Higher algebra" , MIR (1972) (Translated from Russian)</TD></TR></table> | ||
| − | |||
| − | |||
====Comments==== | ====Comments==== | ||
| − | |||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> T.M. Apostol, "Calculus" , '''2''' , Wiley (1969) pp. 93</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> T.M. Apostol, "Calculus" , '''2''' , Wiley (1969) pp. 93</TD></TR></table> | ||
Latest revision as of 17:31, 5 June 2020
If the determinant $ D $
of a square system of linear equations
$$ \begin{array}{c} a _ {11} x _ {1} + \dots + a _ {1n} x _ {n} = b _ {1} , \\ {\dots \dots \dots \dots } \\ a _ {n1} x _ {1} + \dots + a _ {nn} x _ {n} = b _ {n} \end{array} $$
does not vanish, then the system has a unique solution. This solution is given by the formulas
$$ \tag{* } x _ {k} = \ \frac{D _ {k} }{D} ,\ \ k = 1 \dots n. $$
Here $ D _ {k} $ is the determinant obtained from $ D $ when the $ k $- th column is replaced by the column of the free terms $ b _ {1} \dots b _ {n} $. Formulas (*) are known as Cramer's formulas. They have been found by G. Cramer (see [1]).
References
| [1] | G. Cramer, "Introduction à l'analyse des lignes courbes" , Geneva (1750) pp. 657 |
| [2] | A.G. Kurosh, "Higher algebra" , MIR (1972) (Translated from Russian) |
Comments
References
| [a1] | T.M. Apostol, "Calculus" , 2 , Wiley (1969) pp. 93 |
How to Cite This Entry:
Cramer rule. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cramer_rule&oldid=46551
Cramer rule. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cramer_rule&oldid=46551
This article was adapted from an original article by I.V. Proskuryakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article