Difference between revisions of "Cramer's Rule"
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== General Form for n variables == | == General Form for n variables == | ||
− | Cramer's Rule employs the [http:// | + | 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( | We wish to solve the general linear system <math>A \mathbf{x}= \mathbf{b}</math> for the vector <math>\mathbf{x} = \left( |
Revision as of 20:04, 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 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(
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:
Finally, we solve the system: