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 3-D case, the Vector analog of Cramer's rule is
|
(11) |
See also Determinant, Linear Algebra, Matrix, System of Equations, Vector
© 1996-9 Eric W. Weisstein
1999-05-25