## 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)