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
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 3D case, the Vector analog of Cramer's rule is

(11) 
See also Determinant, Linear Algebra, Matrix, System of Equations, Vector
© 19969 Eric W. Weisstein
19990525