Difference between revisions of "Cramer's Rule"

(General Form for n variables)
m (Changed variables to improve readability)
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== General Solution for 2 Variables ==
 
== 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>a, b, c, d, r, s</math>
  
 
<cmath>\begin{eqnarray*}
 
<cmath>\begin{eqnarray*}
x_1x + y_1y &=& a\
+
ax + cy &=& r\
x_2x + y_2y &=& b
+
bx + dy &=& s
 
\end{eqnarray*}</cmath>
 
\end{eqnarray*}</cmath>
  
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<math>x = \frac{\begin{vmatrix}  
 
<math>x = \frac{\begin{vmatrix}  
a & y_1 \
+
r & c \
b & y_2 \end{vmatrix}}
+
s & d \end{vmatrix}}
 
{\begin{vmatrix}  
 
{\begin{vmatrix}  
x_1 & y_1 \  
+
a & c \  
x_2 & y_2 \end{vmatrix}} = \frac{ay_2 - by_1}{x_1y_2 - x_2y_1} \qquad y = \frac{\begin{vmatrix}  
+
b & d \end{vmatrix}} = \frac{rd - sc}{ad - bc} \qquad y = \frac{\begin{vmatrix}  
x_1 & a \
+
a & r \
x_2 & b \end{vmatrix}}
+
b & s \end{vmatrix}}
 
{\begin{vmatrix}  
 
{\begin{vmatrix}  
x_1 & y_1 \  
+
a & c \  
x_2 & y_2 \end{vmatrix}} = \frac{bx_1 - ax_2}{x_1y_2 - y_1x_2}</math>
+
b & d \end{vmatrix}} = \frac{sa - rb}{ad - cb}</math>
  
 
== Example in 3 Variables ==
 
== Example in 3 Variables ==

Revision as of 20:07, 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 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

Given a system of two equations with constants $a, b, c, d, r, s$

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

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

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

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, $A = \left( 123312231 \right) \qquad \mathbf{b} = \left( 141111 \right)$ (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: \[|A| = 18 \qquad |M_1| = 18 \qquad |M_2| = 36 \qquad |M_3| = 54\]

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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