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Cramer's Rule

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

General Form for n variables

Cramer's Rule employs the matrix determinant 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

Consider the following system of linear equations in x and y, with constants a, b, c, d, r, s:

\begin{eqnarray*}ax + cy &=& r\\bx + dy &=& s\end{eqnarray*}

By Cramer's Rule, the solution to this system is:

x = \frac{\begin{vmatrix} r & c \\s & d \end{vmatrix}}{\begin{vmatrix} a & c \\ b & d \end{vmatrix}} = \frac{...

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 \m...

Thus, M_1 = \left( \begin{array}{ccc} 14 & 2 & 3 & 11 & 1 & 2 & 11 & 3 & 1 \end{array} \right) \qqu...

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|} =...

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