Cramer's Rule: Difference between revisions
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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:
