Cramer rule
From HandWiki
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 |
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