Difference between revisions of "Cramer's Rule"
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Let <math>M_j</math> be the matrix formed by replacing the jth column of <math>A</math> with <math>\mathbf{b}</math>. | Let <math>M_j</math> be the matrix formed by replacing the jth column of <math>A</math> with <math>\mathbf{b}</math>. | ||
− | Then, Cramer's Rule states that the general solution is <math>x_j = \frac{|M_j|}{A} \; \; \; \forall j \in \mathbb{N}^{\leq n}</math> | + | Then, Cramer's Rule states that the general solution is <math>x_j = \frac{|M_j|}{|A|} \; \; \; \forall j \in \mathbb{N}^{\leq n}</math> |
== General Solution for 2 Variables == | == General Solution for 2 Variables == | ||
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Finally, we solve the system: | Finally, we solve the system: | ||
<cmath>x_1 = \frac{|M_1|}{|A|} = \frac{18}{18}=1 \qquad x_2 = \frac{|M_2|}{|A|} = \frac{36}{18} = 2 \qquad x_3 = \frac{|M_3|}{|A|} = \frac{54}{18} = 3</cmath> | <cmath>x_1 = \frac{|M_1|}{|A|} = \frac{18}{18}=1 \qquad x_2 = \frac{|M_2|}{|A|} = \frac{36}{18} = 2 \qquad x_3 = \frac{|M_3|}{|A|} = \frac{54}{18} = 3</cmath> | ||
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[[Category:Elementary algebra]] | [[Category:Elementary algebra]] | ||
[[Category:Theorems]] | [[Category:Theorems]] |
Revision as of 08:18, 25 April 2009
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, $A = \left(
Thus,
\[M_1 = \left( \begin{array}{ccc} 14 & 2 & 3 & 11 & 1 & 2 & 11 & 3 & 1 \end{array} \right) \qquad M_2 = \left( \begin{array}{ccc} 1 & 14 & 3 & 3 & 11 & 2 & 2 & 11 & 1 \end{array} \right) \qquad M_3 = \left( \begin{array}{ccc} 1 & 2 & 14 & 3 & 1 & 11 & 2 & 3 & 11 \end{array} \right)\] (Error compiling LaTeX. Unknown error_msg)
We calculate the determinants:
Finally, we solve the system: