Difference between revisions of "Cramer's Rule"

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: $x_1 = \frac{|M_1|}{|A|} = \frac{18}{18}=1 \qquad x_2 = \frac{|M_2|}{|A|} = \frac{36}{18} = 2 \qquad x_3 = \frac{|M_3|}{|A|} = \frac{54}{18} = 3$

@@ Line 1: / Line 1: @@
 '''Cramer's Rule''' is a method of solving systems of equations using [[matrix|matrices]].
-== 2 and 3 Dimensions ==
+== General Form for n variables ==
-Given a system of two equations with constants <math>x_1, x_2, y_1, y_2, a, b</math>
+Cramer's Rule employs the [http://en.wikipedia.org/wiki/Determinant matrix determinant] 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 ==
+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*}
-x_1x + y_1y &=& a\\
+ax + cy &=& r\\
-x_2x + y_2y &=& b
+bx + dy &=& s
 \end{eqnarray*}</cmath>
-Cramer's Rule states that <math>x</math> and <math>y</math> can be found through [[determinant]]s according to the following:
+By Cramer's Rule, the solution to this system is:
-<cmath>\begin{eqnarray*}
+<math>x = \frac{\begin{vmatrix}
-x &=& \frac{\begin{vmatrix}
+r & c \\
-a & y_1 \\
+s & d \end{vmatrix}}
-b & y_2 \end{vmatrix}}
 {\begin{vmatrix}
-x_1 & y_1 \\
+a & c \\
-x_2 & y_2 \end{vmatrix}}\\
+b & d \end{vmatrix}} = \frac{rd - sc}{ad - bc} \qquad y = \frac{\begin{vmatrix}
-y &=& \frac{\begin{vmatrix}
+a & r \\
-x_1 & a \\
+b & s \end{vmatrix}}
-x_2 & b \end{vmatrix}}
 {\begin{vmatrix}
-x_1 & y_1 \\
+a & c \\
-x_2 & y_2 \end{vmatrix}}
+b & d \end{vmatrix}} = \frac{sa - rb}{ad - cb}</math>
-\end{eqnarray*}
-</cmath>
+== Example in 3 Variables ==
+<cmath>\begin{eqnarray*}
+x_1+2x_2+3x_3&=&14\\
+x_1+x_2+2x_3&=&11\\
+x_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>
-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>.
+We calculate the determinants:
+<cmath>|A| = 18 \qquad |M_1| = 18 \qquad |M_2| = 36 \qquad |M_3| = 54</cmath>
-A similar rule is true for 3 by 3 matrices:
+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>
-[[Category:Linear Algebra]]
+[[Category:Algebra]]
+[[Category:Linear algebra]]
+[[Category:Theorems]]

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Difference between revisions of "Cramer's Rule"

Latest revision as of 17:22, 11 October 2023

General Form for n variables

General Solution for 2 Variables

Example in 3 Variables