Difference between revisions of "Cramer's Rule"

(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.
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'''Cramer's Rule''' is a method of solving systems of equations using [[matrix|matrices]].
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== 2 and 3 Dimensions ==
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Given a system of two equations with constants <math>x_1, x_2, y_1, y_2, a, b</math>
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<cmath>\begin{eqnarray*}
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x_1x + y_1y &=& a\
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x_2x + y_2y &=& b
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\end{eqnarray*}</cmath>
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Cramer's Rule states that <math>x</math> and <math>y</math> can be found through [[determinant]]s according to the following:
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<cmath>\begin{eqnarray*}
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x &=& \frac{\begin{vmatrix}
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a & y_1 \
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b & y_2 \end{vmatrix}}
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{\begin{vmatrix}
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x_1 & y_1 \
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x_2 & y_2 \end{vmatrix}}\
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y &=& \frac{\begin{vmatrix}
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x_1 & a \
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x_2 & b \end{vmatrix}}
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{\begin{vmatrix}
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x_1 & y_1 \
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x_2 & y_2 \end{vmatrix}}
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\end{eqnarray*}
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</cmath>
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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>.
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A similar rule is true for 3 by 3 matrices:
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[[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 $x_1, x_2, y_1, y_2, a, b$

\begin{eqnarray*} x_1x + y_1y &=& a\\ x_2x + y_2y &=& b \end{eqnarray*}

Cramer's Rule states that $x$ and $y$ can be found through determinants according to the following:

\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*}

By the rules of determinants, this means that $x = \frac{ay_2 - by_1}{x_1y_2 - x_2y_1}$ and $y = \frac{bx_1 - ax_2}{x_1y_2 - y_1x_2}$.

A similar rule is true for 3 by 3 matrices: