Cramer's rule

Cramer's rule is a theorem in linear algebra, which gives the solution of a system of linear equations in terms of determinants. It is named after Gabriel Cramer (1704 - 1752).

Elementary formulation

The system of equations is represented in matrix multiplication form as:

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:Ax = c

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

where the square matrix A is invertible and the vector x is the column vector of the variables: ( x_i ).

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The theorem then states that:

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:x_i = { det(A_i) over det(A)}

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where A_i is the matrix formed by replacing the ith column of A by the column vector c.

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