Cramer's Rule

Cramer's rule is a method of solving $n$ simultaneous equations for $n$ unknowns.

• Any system of equations of this kind can be written in the form
  

  $$\begin{array}{c} a_{11} \, x_1 + a_{12} \, x_2 + ... + a_{1n... ... x_1 + a_{n2} \, x_n + ... + a_{nn} \, x_n = b_{n} \end{array}$$ (1)

  
  

• The coefficients on the left side of Eq.'s 1 can be written as a matrix,
  

  $$A = \left( \begin{array}{cccc} a_{11} & a_{12} & ... & a_{1... ...& \vdots \\ a_{n1} & a_{n2} & ... & a_{nn} \end{array}\right)$$ (2)

  
  

• The value of any of the $x_i$ can be found via
  

  $$x_i = \frac{\vert B_i\vert}{\vert A\vert}$$ (3)

  
  
  where the notation $\vert A\vert$ and $\vert B_i\vert$ denote the determinants of matrices $A$ and $B_i$, and the matrix $B_i$ is obtained by replacing the $i^{th}$ column of matrix $A$ with the coefficients on the left side of Eq.'s 1. For example,
  

  $$B_2 = \left( \begin{array}{cccc} a_{11} & b_1 & ... & a_{1n... ...ts & \vdots \\ a_{n1} & b_n & ... & a_{nn} \end{array}\right)$$ (4)

  
  

The $\mathbf{n=2}$ Case

• Two equations involving two unknowns can be written in the form
  

  $$\begin{array}{c} a_{11} \, x_1 + a_{12} \, x_2 = b_{1} \\ a_{21} \, x_1 + a_{22} \, x_2 = b_{2} \end{array}$$ (5)

  
  
  which yields
  

  $$A = \left( \begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right)$$ (6)

  
  

• Eq. 3 gives
  

  $$x_1 = \frac{ \left\vert \begin{array}{cc} b_1 & a_{12} \\ ... ... a_{22} - a_{12} \, b_2} {a_{11} \, a_{22} - a_{12} \, a_{21}}$$ (7)

  
  
  and
  

  $$x_2 = \frac{ \left\vert \begin{array}{cc} a_{11} & b_1 \\ a... ... \, b_2 - b_1 \, a_{21}} {a_{11} \, a_{22} - a_{12} \, a_{21}}$$ (8)

  
  

An $\mathbf{n=2}$ Example

• The system of equations
  

  $$\begin{array}{c} 2 \, x_1 - \, x_2 = 1 \\ 5 \, x_1 + 3 \, x_2 = 2 \end{array}$$ (9)

  
  
  has solutions
  

  $$x_1 = \frac{ \left\vert \begin{array}{cc} 1 & -1 \\ 2 & 3 ... ...{cc} 2 & -1 \\ 5 & 3 \end{array}\right\vert } = \frac{5}{11}$$ (10)

  
  
  and
  

  $$x_2 = \frac{ \left\vert \begin{array}{cc} 2 & 1 \\ 5 & 2 ... ...cc} 2 & -1 \\ 5 & 3 \end{array}\right\vert } = -\frac{1}{11}$$ (11)

  
  

• The results can be verified by substituting them into either of Eq.'s 9,
  

  $$\frac{10}{11} + \frac{1}{11} = \frac{11}{11} = 1 \hspace{24pt}\checkmark$$ (12)

  
  

An $\mathbf{n=3}$ Example

• The system of equations
  

  $$\begin{array}{c} 2 \, x_1 - ~ \, x_2 + 4 \, x_3 = 2 \\ 5 \,... ... = 1 \\ ~~~ \, x_1 + 6 \, x_2 + ~~ \, x_3 = -3 \\ \end{array}$$ (13)

  
  
  has solutions
  

  $$x_1 = \frac{ \left\vert \begin{array}{ccc} 2 & -1 & 4 \\ 1 & 3 & 2 \\ ... ...gin{array}{ccc} 2 & -1 & 4 \\ 5 & 3 & 2 \\ 1 & 6 & 1 \end{array}\right\vert }$$

  $$= \frac{ 2 (3 \cdot 1 - 2 \cdot 6) - (-1)(1 \cdot 1 - 2 \cdot -3) +... ...cdot 1 - 2 \cdot 6) - (-1)(5 \cdot 1 - 2 \cdot 1) + 4 (5 \cdot 6 - 3 \cdot 1) }$$

  $$= \frac{49}{93}$$ (14)

  
  
  
  

  $$x_2 = \frac{ \left\vert \begin{array}{ccc} 2 & 2 & 4 \\ 5 & 1 & 2 \\ ... ...gin{array}{ccc} 2 & -1 & 4 \\ 5 & 3 & 2 \\ 1 & 6 & 1 \end{array}\right\vert }$$

  $$= \frac{ 2 (1 \cdot 1 - 2 \cdot -3) - 2 (5 \cdot 1 - 2 \cdot 1) + 4 (5 \cdot -3 - 1 \cdot 1) }{93}$$

  $$= -\frac{56}{93}$$ (15)

  
  
  and
  

  $$x_3 = \frac{ \left\vert \begin{array}{ccc} 2 & -1 & 2 \\ 5 & 3 & 1 \\ ... ...gin{array}{ccc} 2 & -1 & 4 \\ 5 & 3 & 2 \\ 1 & 6 & 1 \end{array}\right\vert }$$

  $$= \frac{ 2 (3 \cdot -3 - 1 \cdot 6) - (-1)(5 \cdot -3 - 1 \cdot 1) + 2 (5 \cdot 6 - 3 \cdot 1) } {93}$$

  $$= \frac{8}{93}$$ (16)

  
  

• The results can be verified by substituting them into any of Eq.'s 13,
  

  $$\frac{2 \cdot 49 - (-56) + 4 \cdot 8}{93} = \frac{186}{93} = 2 \hspace{24pt}\checkmark$$ (17)

  
  

Copyright © 2002-2004, Lewis A. Riley

Updated Mon Jan 19 13:29:10 2004

This work is licensed under a Creative Commons License.