# How do I solve a system of 3 equations with a matrix?

Dec 25, 2014

I would use Cramer's Rule .
This rule is based upon using the matrix associated with the coefficients of the unknowns and of the pure coefficients of your system, evaluate their determinants and combine them to evaluate the values of the unknowns.

Consider an example:

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 column of pure coefficients (the ones without unknown):

We evaluate the three determinants for these matrices:

Finally we can calculate the values of the unknowns as:

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

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