Given a set of linear equations
(1)
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consider the determinant
(2)
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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)
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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)
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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)
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and similarly for
(6)
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(7)
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This procedure can be generalized to a set of equations so, given a system of linear equations
(8)
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let
(9)
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If , then nondegenerate solutions exist only if . If and , the system has no unique solution. Otherwise, compute
(10)
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Then for . In the three-dimensional case, the vector analog of Cramer's rule is
(11)
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