Consider the system of three equations shown below: $$ a_{11}x_1 + a_{12}x_2 + a_{13}x_3 = a_{14} $$ $$ a_{21}x_1 + a_{22}x_2 + a_{23}x_3 = a_{24} $$ $$ a_{31}x_1 + a_{32}x_2 + a_{33}x_3 = a_{34} $$ ...which we use to create the following augmented matrix of coefficients: $$ \begin{pmatrix} a_{11}&a_{12}&a_{13}&a_{14}\\ a_{21}&a_{22}&a_{23}&a_{24}\\ a_{31}&a_{32}&a_{33}&a_{34}\\ \end{pmatrix} $$ ...and the following matrix of coefficients: $$ \begin{pmatrix} a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33}\\ \end{pmatrix} $$ Our Determinant (D) is the following: $$ D = \begin{vmatrix} a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33}\\ \end{vmatrix} $$ $$ = a_{11} \begin{vmatrix} a_{22}&a_{23}\\ a_{32}&a_{33}\\ \end{vmatrix} - a_{12} \begin{vmatrix} a_{21}&a_{23}\\ a_{31}&a_{33}\\ \end{vmatrix} + a_{13} \begin{vmatrix} a_{21}&a_{22}\\ a_{31}&a_{32}\\ \end{vmatrix} $$ Don't forget to alternate signs when calculating determinants. $$ D = a_{11}(a_{22}a_{33}-a_{32}a_{23}) $$ $$ \qquad - a_{12}(a_{21}a_{33}-a_{31}a_{23}) $$ $$ \qquad + a_{13}(a_{21}a_{32}-a_{22}a_{31}) $$

We now proceed to calculate the following determinants. Start by creating a matrix that consists of the matrix of coefficients but with the first column replaced by the 4th column of the augmented matrix of coefficients: $$ D_{x1} = \begin{vmatrix} a_{14}&a_{12}&a_{13}\\ a_{24}&a_{22}&a_{23}\\ a_{34}&a_{32}&a_{33}\\ \end{vmatrix} $$ Now calculate the determinant of a matrix consisting of the matrix of coefficients with the 2nd column replaced by the 4th column of the augmented matrix of coefficients: $$ D_{x2} = \begin{vmatrix} a_{11}&a_{14}&a_{13}\\ a_{21}&a_{24}&a_{23}\\ a_{31}&a_{34}&a_{33}\\ \end{vmatrix} $$ Finally calculate the determinant of a matrix consisting of the matrix of coefficients where the 3rd column is replaced by the 4th column of the augmented matrix of coefficients: $$ D_{x3} = \begin{vmatrix} a_{11}&a_{12}&a_{14}\\ a_{21}&a_{22}&a_{24}\\ a_{31}&a_{32}&a_{34}\\ \end{vmatrix} $$

Cramer's Rule:

Cramer's Rule states that the solutions to the system of equations given at the top of this page is as follows: $$ x_1 = \frac{D_{x1}}{D} $$ $$ x_2 = \frac{D_{x2}}{D} $$ $$ x_3 = \frac{D_{x3}}{D} $$

Let's take a look at an example:

Continue on to -Cramer's Rule (example problem)