# 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)