Difference between revisions of "Cramer's Rule"
Starcellist (talk | contribs) (New page: Cramer's Rule is a method of solving systems of equations using matrices.) |
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| − | Cramer's Rule is a method of solving systems of equations using matrices. | + | '''Cramer's Rule''' is a method of solving systems of equations using [[matrix|matrices]]. |
| + | |||
| + | == 2 and 3 Dimensions == | ||
| + | Given a system of two equations with constants <math>x_1, x_2, y_1, y_2, a, b</math> | ||
| + | |||
| + | <cmath>\begin{eqnarray*} | ||
| + | x_1x + y_1y &=& a\\ | ||
| + | x_2x + y_2y &=& b | ||
| + | \end{eqnarray*}</cmath> | ||
| + | |||
| + | Cramer's Rule states that <math>x</math> and <math>y</math> can be found through [[determinant]]s according to the following: | ||
| + | |||
| + | <cmath>\begin{eqnarray*} | ||
| + | x &=& \frac{\begin{vmatrix} | ||
| + | a & y_1 \\ | ||
| + | b & y_2 \end{vmatrix}} | ||
| + | {\begin{vmatrix} | ||
| + | x_1 & y_1 \\ | ||
| + | x_2 & y_2 \end{vmatrix}}\\ | ||
| + | y &=& \frac{\begin{vmatrix} | ||
| + | x_1 & a \\ | ||
| + | x_2 & b \end{vmatrix}} | ||
| + | {\begin{vmatrix} | ||
| + | x_1 & y_1 \\ | ||
| + | x_2 & y_2 \end{vmatrix}} | ||
| + | \end{eqnarray*} | ||
| + | </cmath> | ||
| + | |||
| + | By the rules of determinants, this means that <math>x = \frac{ay_2 - by_1}{x_1y_2 - x_2y_1}</math> and <math>y = \frac{bx_1 - ax_2}{x_1y_2 - y_1x_2}</math>. | ||
| + | |||
| + | A similar rule is true for 3 by 3 matrices: | ||
| + | |||
| + | [[Category:Linear Algebra]] | ||
Revision as of 17:29, 23 October 2007
Cramer's Rule is a method of solving systems of equations using matrices.
2 and 3 Dimensions
Given a system of two equations with constants 
Cramer's Rule states that 
and 
can be found through determinants according to the following:
By the rules of determinants, this means that 
and 
.
A similar rule is true for 3 by 3 matrices:
