Cramer's Rule

Cramer's Rule is a method of solving systems of equations using matrices.

2 and 3 Dimensions

Given a system of two equations with constants $x_1, x_2, y_1, y_2, a, b$

$\begin{eqnarray*} x_1x + y_1y &=& a\\ x_2x + y_2y &=& b \end{eqnarray*}$

Cramer's Rule states that $x$ and $y$ can be found through determinants according to the following:

$\begin{eqnarray*} x &=& \frac{\begin{vmatrix} a & y_1 \\ b & y_2 \end{vmatrix}} {\begin{vmatrix} x_1 & y_1 \\ x_2 & y_2 \end{vmatrix}}\\ y &=& \frac{\begin{vmatrix} x_1 & a \\ x_2 & b \end{vmatrix}} {\begin{vmatrix} x_1 & y_1 \\ x_2 & y_2 \end{vmatrix}} \end{eqnarray*}$

By the rules of determinants, this means that $x = \frac{ay_2 - by_1}{x_1y_2 - x_2y_1}$ and $y = \frac{bx_1 - ax_2}{x_1y_2 - y_1x_2}$ .

A similar rule is true for 3 by 3 matrices:

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Cramer's Rule

2 and 3 Dimensions