General Case
Consider a system of n linear equations for n unknowns, represented in matrix multiplication form as follows:
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where the n by n matrix has a nonzero determinant, and the vector is the column vector of the variables.
Then the theorem states that in this case the system has a unique solution, whose individual values for the unknowns are given by:
where is the matrix formed by replacing the ith column of by the column vector .
The rule holds for systems of equations with coefficients and unknowns in any field, not just in the real numbers. It has recently been shown that Cramer's rule can be implemented in O(n3) time, which is comparable to more common methods of solving systems of linear equations, such as Gaussian elimination.
Read more about this topic: Cramer's Rule
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