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# 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 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

 (11)

Determinant, Linear Algebra, Matrix, System of Equations, Vector

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## References

Cramer, G. "Intr. à l'analyse de lignes courbes algébriques." Geneva, 657-659, 1750.Muir, T. The Theory of Determinants in the Historical Order of Development, Vol. 1. New York: Dover, pp. 11-14, 1960.

Cramer's Rule

## Cite this as:

Weisstein, Eric W. "Cramer's Rule." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CramersRule.html