Difference between revisions of "Cramer's Rule"
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'''Cramer's Rule''' is a method of solving systems of equations using [[matrix|matrices]]. | '''Cramer's Rule''' is a method of solving systems of equations using [[matrix|matrices]]. | ||
| − | == 2 | + | == General Form for n variables == |
| + | Cramer's Rule employs the [http://www.example.com link title] to solve a system of ''n'' linear equations in ''n'' variables. | ||
| + | |||
| + | We wish to solve the general linear system <math>A \mathbf{x}= \mathbf{b}</math> for the vector <math>\mathbf{x} = \left( \begin{array}{c} x_1 \\ \vdots \\ x_n \end{array} \right)</math>. Here, <math>A</math> is the coefficient matrix, <math>\mathbf{b}</math> is a column vector. | ||
| + | |||
| + | 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> | ||
| + | |||
| + | == General Solution for 2 Variables == | ||
Given a system of two equations with constants <math>x_1, x_2, y_1, y_2, a, b</math> | Given a system of two equations with constants <math>x_1, x_2, y_1, y_2, a, b</math> | ||
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\end{eqnarray*}</cmath> | \end{eqnarray*}</cmath> | ||
| − | Cramer's Rule states that <math>x</math> and <math>y</math> can be found through | + | Cramer's Rule states that <math>x</math> and <math>y</math> can be found through determinants according to the following: |
| − | < | + | <math>x = \frac{\begin{vmatrix} |
| − | x | ||
a & y_1 \\ | a & y_1 \\ | ||
b & y_2 \end{vmatrix}} | b & y_2 \end{vmatrix}} | ||
{\begin{vmatrix} | {\begin{vmatrix} | ||
x_1 & y_1 \\ | x_1 & y_1 \\ | ||
| − | x_2 & y_2 \end{vmatrix}}\\ | + | x_2 & y_2 \end{vmatrix}} = \frac{ay_2 - by_1}{x_1y_2 - x_2y_1} \qquad y = \frac{\begin{vmatrix} |
| − | y | ||
x_1 & a \\ | x_1 & a \\ | ||
x_2 & b \end{vmatrix}} | x_2 & b \end{vmatrix}} | ||
{\begin{vmatrix} | {\begin{vmatrix} | ||
x_1 & y_1 \\ | x_1 & y_1 \\ | ||
| − | x_2 & y_2 \end{vmatrix}} | + | x_2 & y_2 \end{vmatrix}} = \frac{bx_1 - ax_2}{x_1y_2 - y_1x_2}</math> |
| − | \end{eqnarray*} | + | |
| − | </cmath> | + | == Example in 3 Variables == |
| + | |||
| + | <cmath>\begin{eqnarray*} | ||
| + | x_1+2x_2+3x_3&=&14\\ | ||
| + | 3x_1+x_2+2x_3&=&11\\ | ||
| + | 2x_1+3x_2+x_3&=&11 | ||
| + | \end{eqnarray*}</cmath> | ||
| + | |||
| + | 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 & 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: | |
| + | <cmath>|A| = 18 \qquad |M_1| = 18 \qquad |M_2| = 36 \qquad |M_3| = 54</cmath> | ||
| − | A | + | 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> | ||
{{incomplete|article}} | {{incomplete|article}} | ||
Revision as of 21:03, 19 May 2008
Cramer's Rule is a method of solving systems of equations using matrices.
General Form for n variables
Cramer's Rule employs the link title 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
Given a system of two equations with constants 
Cramer's Rule states that 
and 
can be found through determinants according to the following:
Example in 3 Variables
Here, $A = \left( \begin{array}{ccc} 1 & 2 & 3 & 3 & 1 & 2 & 2 & 3 & 1 \end{array} \right) \qquad \mathbf{b} = \left( \begin{array}{c} 14 & 11 & 11 \end{array} \right)$ (Error compiling LaTeX. Unknown error_msg)
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:
![\[|A| = 18 \qquad |M_1| = 18 \qquad |M_2| = 36 \qquad |M_3| = 54\]](http://latex.artofproblemsolving.com/7/1/d/71d2b0cb5acd430ed01cf5b2ccd4df8984eaba62.png)
Finally, we solve the system:
![\[x_1 = \frac{|M_1|}{|A|} = \frac{18}{18}=1 \qquad x_2 = \frac{|M_2|}{|A|} = \frac{36}{18} = 2 \qquad x_3 = \frac{|M_3|}{|A|} = \frac{54}{18} = 3\]](http://latex.artofproblemsolving.com/e/2/3/e23518ccee04b1be493b55b9140594f032a93fe7.png)
