Cramer's rule is an algorithm for solving square linear systems using determinants of matrices.^{[2]} Cramer's rule can only be used with linear systems that have exactly one solution. Cramer's rule is named after Gabriel Cramer (1704–52), a Swiss mathematician.
Cramer's rule uses determinants to find the solution of a linear system. For example, start with the linear system
Now convert this linear system into a 3x4 matrix. We will be making four determinants out of this matrix. Each of these determinants will be based on a 3x3 square matrix. The first determinant, which we will label |A_{0}|, is the first three columns of the matrix. Now find the value of |A_{0}|. The second determinant will be based on the first. Substitute the last column of matrix A into the first column of determinant |A_{0}|. The new determinant is labeled |A_{x}|. Now find the value of |A_{x}|. Continue by finding the determinate of |A_{y}|. Do this by substituting the last column of matrix A into the second column of determinant |A_{0}|. Now repeat the pattern for |A_{z}|. The solution for the linear system is So# | A | B | C | D |
E | F | G | H | I |
J | K | L | M | N |
O | P | Q | R | S |
T | U | V | W | X |
Y | Z |
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