# Difference between revisions of "Cramer's Rule"

5849206328x (talk | contribs) m |
Mathgeek2006 (talk | contribs) m (→Example in 3 Variables) |
||

Line 40: | Line 40: | ||

\end{eqnarray*}</cmath> | \end{eqnarray*}</cmath> | ||

− | Here, <math>A = \left( \begin{array}{ccc} 1 & 2 & 3 | + | Here, <math>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)</math> |

− | Thus, <cmath>M_1 = \left( \begin{array}{ccc} 14 & 2 & 3 | + | Thus, <cmath>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)</cmath> |

We calculate the determinants: | We calculate the determinants: |

## Latest revision as of 19:09, 10 March 2015

**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 for the vector . Here, is the coefficient matrix, is a column vector.

Let be the matrix formed by replacing the jth column of with .

Then, Cramer's Rule states that the general solution is

## General Solution for 2 Variables

Consider the following system of linear equations in and , with constants :

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

## Example in 3 Variables

Here,

Thus,

We calculate the determinants:

Finally, we solve the system: