Cramer's Rule

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

General Form for n variables

Cramer's Rule employs the link title to solve a system of n linear equations in n variables.

We wish to solve the general linear system $A \mathbf{x}= \mathbf{b}$ for the vector $\mathbf{x} = \left( \begin{array}{c} x_1 \\ \vdots \\ x_n \end{array} \right)$ . Here, $A$ is the coefficient matrix, $\mathbf{b}$ is a column vector.

Let $M_j$ be the matrix formed by replacing the jth column of $A$ with $\mathbf{b}$ .

Then, Cramer's Rule states that the general solution is $x_j = \frac{|M_j|}{A} \; \; \; \forall j \in \mathbb{N}^{\leq n}$

General Solution for 2 Variables

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:

$x = \frac{\begin{vmatrix} a & y_1 \\ b & y_2 \end{vmatrix}} {\begin{vmatrix} x_1 & y_1 \\ x_2 & y_2 \end{vmatrix}} = \frac{ay_2 - by_1}{x_1y_2 - x_2y_1} \qquad y = \frac{\begin{vmatrix} x_1 & a \\ x_2 & b \end{vmatrix}} {\begin{vmatrix} x_1 & y_1 \\ x_2 & y_2 \end{vmatrix}} = \frac{bx_1 - ax_2}{x_1y_2 - y_1x_2}$

Example in 3 Variables

$\begin{eqnarray*} x_1+2x_2+3x_3&=&14\\ 3x_1+x_2+2x_3&=&11\\ 2x_1+3x_2+x_3&=&11 \end{eqnarray*}$

Here, $A = \left( \begin{array}{ccc} 1 & 2 & 3 & 3 & 1 & 2 & 2 & 3 & 1 \end{array} \right) \qquad \mathbf{b} = \left( \begin{array}{c} 14 & 11 & 11 \end{array} \right)$ (Error compiling LaTeX. Unknown error_msg)

Thus,

\[M_1 = \left( \begin{array}{ccc} 14 & 2 & 3 & 11 & 1 & 2 & 11 & 3 & 1 \end{array} \right) \qquad M_2 = \left( \begin{array}{ccc} 1 & 14 & 3 & 3 & 11 & 2 & 2 & 11 & 1 \end{array} \right) \qquad M_3 = \left( \begin{array}{ccc} 1 & 2 & 14 & 3 & 1 & 11 & 2 & 3 & 11 \end{array} \right)\] (Error compiling LaTeX. Unknown error_msg)

We calculate the determinants: $|A| = 18 \qquad |M_1| = 18 \qquad |M_2| = 36 \qquad |M_3| = 54$

Finally, we solve the system: $x_1 = \frac{|M_1|}{|A|} = \frac{18}{18}=1 \qquad x_2 = \frac{|M_2|}{|A|} = \frac{36}{18} = 2 \qquad x_3 = \frac{|M_3|}{|A|} = \frac{54}{18} = 3$

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Cramer's Rule

General Form for n variables

General Solution for 2 Variables

Example in 3 Variables