Cramer’s Rule

Cramer’s Rule uses determinants to solve systems of linear equations . Consider the system of two linear equations in two variables:
$\begin{array}{l} a x + b y = c \\ d x + e y = f \end{array}$

Using the linear combination method, you can verify that

$x = \frac{c e - b f}{a e - b d}$ and $y = \frac{a f - c d}{a e - b d}$ if $a e - b d \neq 0$

Note that the denominators are equal to the determinant of the coefficients.

$D = | \begin{matrix} a & b \\ d & e \end{matrix} |$

The numerators are equal to the determinants $D_{x}$ and $D_{y}$ where

$D_{x} = | \begin{matrix} c & b \\ f & e \end{matrix} |$ and $D_{y} = | \begin{matrix} a & c \\ d & f \end{matrix} |$

$D_{x}$ is formed by replacing the column of coefficients of $x$ in $D$ with the column of constants and $D_{y}$ is formed by replacing the column of coefficients of $y$ in $D$ with the column of constants.

By substitution,

$x = \frac{| \begin{matrix} c & b \\ f & e \end{matrix} |}{| \begin{matrix} a & b \\ d & e \end{matrix} |} = \frac{D_{x}}{D}$ and $y = \frac{| \begin{matrix} a & c \\ d & f \end{matrix} |}{| \begin{matrix} a & b \\ d & e \end{matrix} |} = \frac{D_{y}}{D}$ .

Example:

Use determinants to solve the system of equations: ${\begin{cases} x - 5 y = 2 \\ 2 x + y = 4 \end{cases}$

$\begin{array}{l} x = \frac{D_{x}}{D} = \frac{| \begin{array}{r} 2 & - 5 \\ 4 & 1 \end{array} |}{| \begin{array}{r} 1 & - 5 \\ 2 & 1 \end{array} |} = \frac{2 + 20}{1 + 10} = \frac{22}{11} = 2 \\ y = \frac{D_{y}}{D} = \frac{| \begin{array}{r} 1 & 2 \\ 2 & 4 \end{array} |}{| \begin{array}{r} 1 & - 5 \\ 2 & 1 \end{array} |} = \frac{4 - 4}{1 + 10} = \frac{0}{11} = 0 \end{array}$

Therefore, the solution is $(2, 0)$ .

Determinants can also be used to solve a system of linear equations in three variables:

$\begin{array}{l} a_{1} x + b_{1} y + c_{1} z = d_{1} \\ a_{2} x + b_{2} y + c_{2} z = d_{2} \\ a_{3} x + b_{3} y + c_{3} z = d_{3} \end{array}$

Then,

$\begin{matrix} D = | \begin{matrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{matrix} | \neq 0 & D_{x} = | \begin{matrix} d_{1} & b_{1} & c_{1} \\ d_{2} & b_{2} & c_{2} \\ d_{3} & b_{3} & c_{3} \end{matrix} | \\ D_{y} = | \begin{matrix} a_{1} & d_{1} & c_{1} \\ a_{2} & d_{2} & c_{2} \\ a_{3} & d_{3} & c_{3} \end{matrix} | & D_{z} = | \begin{matrix} a_{1} & b_{1} & d_{1} \\ a_{2} & b_{2} & d_{2} \\ a_{3} & b_{3} & d_{3} \end{matrix} | \end{matrix}$

And,

$x = \frac{D_{x}}{D}, y = \frac{D_{y}}{D}, z = \frac{D_{z}}{D}$ .

This method can be generalized for a system of $n$ linear equations in $n$ variables.

It was named for the Swiss mathematician Gabriel Cramer.

Cramer’s Rule

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