Difference between revisions of "Cramer's Rule"
5849206328x (talk | contribs) m |
Mathgeek2006 (talk | contribs) m (→Example in 3 Variables) |
||
Line 40: | Line 40: | ||
\end{eqnarray*}</cmath> | \end{eqnarray*}</cmath> | ||
− | Here, <math>A = \left( \begin{array}{ccc} 1 & 2 & 3 | + | 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 | + | 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> |
We calculate the determinants: | We calculate the determinants: |
Revision as of 18:09, 10 March 2015
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: