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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]].
  
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== 2 and 3 Dimensions ==
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== General Form for n variables ==
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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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Cramer's Rule employs the [http://en.wikipedia.org/wiki/Determinant matrix determinant] to solve a system of ''n'' linear equations in ''n'' variables.
 +
 
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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.
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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>.
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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>
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== General Solution for 2 Variables ==
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Consider the following system of linear equations in <math>x</math> and <math>y</math>, with constants <math>a, b, c, d, r, s</math>:
  
 
<cmath>\begin{eqnarray*}
 
<cmath>\begin{eqnarray*}
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x_1x + y_1y &=& a\\
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ax + cy &=& r\\
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x_2x + y_2y &=& b
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bx + dy &=& s
 
\end{eqnarray*}</cmath>
 
\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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By Cramer's Rule, the solution to this system is:
  
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<cmath>\begin{eqnarray*}
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<math>x = \frac{\begin{vmatrix}  
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x &=& \frac{\begin{vmatrix}  
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r & c \\
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a & y_1 \\
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s & d \end{vmatrix}}
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b & y_2 \end{vmatrix}}
 
 
{\begin{vmatrix}  
 
{\begin{vmatrix}  
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x_1 & y_1 \\  
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a & c \\  
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x_2 & y_2 \end{vmatrix}}\\
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b & d \end{vmatrix}} = \frac{rd - sc}{ad - bc} \qquad y = \frac{\begin{vmatrix}  
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y &=& \frac{\begin{vmatrix}  
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a & r \\
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x_1 & a \\
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b & s \end{vmatrix}}
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x_2 & b \end{vmatrix}}
 
 
{\begin{vmatrix}  
 
{\begin{vmatrix}  
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x_1 & y_1 \\  
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a & c \\  
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x_2 & y_2 \end{vmatrix}}
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b & d \end{vmatrix}} = \frac{sa - rb}{ad - cb}</math>
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\end{eqnarray*}
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</cmath>
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== Example in 3 Variables ==
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<cmath>\begin{eqnarray*}
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x_1+2x_2+3x_3&=&14\\
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3x_1+x_2+2x_3&=&11\\
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2x_1+3x_2+x_3&=&11
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\end{eqnarray*}</cmath>
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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>
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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>
  
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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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We calculate the determinants:
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<cmath>|A| = 18 \qquad |M_1| = 18 \qquad |M_2| = 36 \qquad |M_3| = 54</cmath>
  
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A similar rule is true for 3 by 3 matrices:
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Finally, we solve the system:
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<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]]
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[[Category:Algebra]]
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[[Category:Linear algebra]]
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[[Category:Theorems]]

Latest revision as of 17:22, 11 October 2023

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 $A \mathbf{x}= \mathbf{b}$ for the vector $\mathbf{x} = \left( \begin{array}{c} x_1 \\ \vdots \\ x_n \end{array} \right)$. Here, $A$ is the coefficient matrix, $\mathbf{b}$ is a column vector.

Let $M_j$ be the matrix formed by replacing the jth column of $A$ with $\mathbf{b}$.

Then, Cramer's Rule states that the general solution is $x_j = \frac{|M_j|}{|A|} \; \; \; \forall j \in \mathbb{N}^{\leq n}$

General Solution for 2 Variables

Consider the following system of linear equations in $x$ and $y$, with constants $a, b, c, d, r, s$:

\begin{eqnarray*} ax + cy &=& r\\ bx + dy &=& s \end{eqnarray*}

By Cramer's Rule, the solution to this system is:

$x = \frac{\begin{vmatrix} r & c \\ s & d \end{vmatrix}} {\begin{vmatrix} a & c \\ b & d \end{vmatrix}} = \frac{rd - sc}{ad - bc} \qquad y = \frac{\begin{vmatrix} a & r \\ b & s \end{vmatrix}} {\begin{vmatrix} a & c \\ b & d \end{vmatrix}} = \frac{sa - rb}{ad - cb}$

Example in 3 Variables

\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*}

Here, $A = \left( \begin{array}{ccc} 1 & 2 & 3\\ 3 & 1 & 2\\ 2 & 3 & 1 \end{array} \right) \qquad \mathbf{b} = \left( \begin{array}{c} 14\\ 11\\ 11 \end{array} \right)$

Thus, \[M_1 = \left( \begin{array}{ccc} 14 & 2 & 3\\ 11 & 1 & 2\\ 11 & 3 & 1 \end{array} \right) \qquad M_2 = \left( \begin{array}{ccc} 1 & 14 & 3\\ 3 & 11 & 2\\ 2 & 11 & 1 \end{array} \right) \qquad M_3 = \left( \begin{array}{ccc} 1 & 2 & 14\\ 3 & 1 & 11\\ 2 & 3 & 11 \end{array} \right)\]

We calculate the determinants: \[|A| = 18 \qquad |M_1| = 18 \qquad |M_2| = 36 \qquad |M_3| = 54\]

Finally, we solve the system: \[x_1 = \frac{|M_1|}{|A|} = \frac{18}{18}=1 \qquad x_2 = \frac{|M_2|}{|A|} = \frac{36}{18} = 2 \qquad x_3 = \frac{|M_3|}{|A|} = \frac{54}{18} = 3\]

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