Cramer rule

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)

References

[a1]	T.M. Apostol, "Calculus" , 2 , Wiley (1969) pp. 93