Difference between revisions of "Cramer's Rule"
(expand, definition) |
(Tag: Undo) |
||
(10 intermediate revisions by 6 users not shown) | |||
Line 1: | Line 1: | ||
'''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]]. | ||
− | == | + | == General Form for n variables == |
− | + | 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*} | <cmath>\begin{eqnarray*} | ||
− | + | ax + cy &=& r\ | |
− | + | bx + dy &=& s | |
\end{eqnarray*}</cmath> | \end{eqnarray*}</cmath> | ||
− | Cramer's Rule | + | By Cramer's Rule, the solution to this system is: |
− | < | + | <math>x = \frac{\begin{vmatrix} |
− | x | + | r & c \ |
− | + | s & d \end{vmatrix}} | |
− | |||
{\begin{vmatrix} | {\begin{vmatrix} | ||
− | + | a & c \ | |
− | + | b & d \end{vmatrix}} = \frac{rd - sc}{ad - bc} \qquad y = \frac{\begin{vmatrix} | |
− | y | + | a & r \ |
− | + | b & s \end{vmatrix}} | |
− | |||
{\begin{vmatrix} | {\begin{vmatrix} | ||
− | + | a & c \ | |
− | + | 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\ | ||
+ | 3x_1+x_2+2x_3&=&11\ | ||
+ | 2x_1+3x_2+x_3&=&11 | ||
+ | \end{eqnarray*}</cmath> | ||
+ | |||
+ | Here, <math>A = \left( | ||
+ | |||
+ | Thus, <cmath>M_1 = \left( | ||
− | + | 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> | ||
− | [[Category:Linear | + | [[Category:Algebra]] |
+ | [[Category:Linear algebra]] | ||
+ | [[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 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,
Thus,
We calculate the determinants:
Finally, we solve the system: