# Cramer’s Rule

## Cramer’s Rule

Method that can be used to determine the solution to a system of n linear equations in n variables whose determinant is not equal to zero.

Cramer’s rule uses the concept of determinant of a matrix.
As a general rule, given the equations $$A_{1}$$x + $$B_{1}$$y = $$C_{1}$$ and $$A_{2}$$x + $$B_{2}$$y = $$C_{2}$$, then :

D = $$\begin{pmatrix}A_{1} & B_{1}\\A_{2} & B_{2}\end{pmatrix}$$ = A$$_{1}$$× B$$_{2}$$ – A$$_{2}$$× B$$_{1}$$

D$$_{1}$$ = $$\begin{pmatrix}C_{1} & B_{1}\\C_{2} & B_{2}\end{pmatrix}$$ = C$$_{1}$$× B$$_{2}$$ – B$$_{1}$$× C$$_{2}$$

D$$_{2}$$ = $$\begin{pmatrix}A_{1} & C_{1}\\A_{2} & C_{2}\end{pmatrix}$$ = A$$_{1}$$× C$$_{2}$$ – A$$_{2}$$× C$$_{1}$$

x = $$\dfrac{D_{1}}{D}$$ where D ≠ 0 and  y = $$\dfrac{D_{2}}{D}$$ where D ≠ 0

### Example

Given the equations 3x + 2y = 12 and 5x + 2y = 16, then :

D = $$\begin{pmatrix}3 & 2\\5 & 2\end{pmatrix}$$ = 3 × 2 – 5 × 2 = –4

D$$_{1}$$ = $$\begin{pmatrix}12 & 2\\16 & 2\end{pmatrix}$$ = 12 × 2 – 2 × 16 = –8

D$$_{2}$$ = $$\begin{pmatrix}3 & 12\\5 & 16\end{pmatrix}$$ = 3 × 16 – 5 × 12 = –12

x = $$\dfrac{–8}{–4}$$ = 2  and y = $$\dfrac{–12}{–4}$$ = 3

### Historical note*

Gabriel Cramer (1704-1752) was a Swiss mathematician who, after attempting to solve a system of five equations in five unknowns using the general elimination method, wrote in his work Introduction à l’analyse des lignes courbes algébriques (Introduction to the Analysis of Algebraic Curves) : “I believe I may have found a convenient and general rule for any given number of equations and unknowns that do not have a degree higher than one … ” [Translation] Although Cramer came to be known mainly for this rule, credit rightly belongs to Colin Maclaurin (1698-1746), a Scottish mathematician who described the rule in his Treatise of Algebra, which was published in 1748, two years after his death.

*Source: Jean-Paul Collette, Histoire des mathématiques, Volume 2, pp. 101-104.