Now multiply
by , and use the property of determinants
that multiplication by a constant is equivalent
to multiplication of each entry in a single column
by that constant, so

(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 three-dimensional
case, the vector analog of Cramer's rule is