Difference between revisions of "Cramer's Rule"
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== General Solution for 2 Variables == | == 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*} | ||
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\end{eqnarray*}</cmath> | \end{eqnarray*}</cmath> | ||
− | Cramer's Rule | + | By Cramer's Rule, the solution to this system is: |
<math>x = \frac{\begin{vmatrix} | <math>x = \frac{\begin{vmatrix} |
Revision as of 20:08, 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
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, $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: