# What is Cramer's rule?

Dec 26, 2014

Cramer's Rule.
This rule is based upon manipulation of determinants of the matrices associated with the numerical coefficients of your system.

You just pick the variable you want to solve for, replace that variable's column of values in the coefficient determinant with the answer-column's values, evaluate that determinant, and divide by the coefficient determinant.

It works with systems with a number of equations equal to the number of unknowns. it also works well up to systems of 3 equations in 3 unknowns. More than that and you´ll have better chances using reductions methods (row echelon form).

Consider an example:

(NOTE: if $\det \left(A\right) = 0$ you cannot use Cramer's Rule and your system will not have a unique solution).

Now we consider 3 other matrices, ${A}_{x} , {A}_{y} \mathmr{and} {A}_{z}$ and their determinant. These matrices are obtained by substituting each column of $A$ with the answer-column's values (the ones without unknown):

We evaluate the three determinants for these matrices:

Finally we can calculate the values of the unknowns as:

$x = \det \frac{{A}_{x}}{\det \left(A\right)} = \frac{- 60}{-} 60 = 1$
$y = \det \frac{{A}_{y}}{\det \left(A\right)} = \frac{- 240}{-} 60 = 4$
$z = \det \frac{{A}_{z}}{\det \left(A\right)} = \frac{120}{-} 60 = - 2$

$x = 1$
$y = 4$
$z = - 2$