## Cramer's Rule

Given a set of linear equations (1)

consider the Determinant (2)

Now multiply by , and use the property of Determinants that Multiplication by a constant is equivalent to Multiplication of each entry in a given row by that constant (3)

Another property of Determinants enables us to add a constant times any column to any column and obtain the same Determinant, so add times column 2 and times column 3 to column 1, (4)

If , then (4) reduces to , so the system has nondegenerate solutions (i.e., solutions other than (0, 0, 0)) only if (in which case there is a family of solutions). If and , the system has no unique solution. If instead and , then solutions are given by (5)

and similarly for   (6)   (7)

This procedure can be generalized to a set of equations so, given a system of linear equations (8)

let (9)

If , then nondegenerate solutions exist only if . If and , the system has no unique solution. Otherwise, compute (10)

Then for . In the 3-D case, the Vector analog of Cramer's rule is (11)