In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the vector of right hand sides of the equations. It is named after Gabriel Cramer (1704–1752), who published the rule for an arbitrary numbers of unknowns in 1750, although Colin Maclaurin also published special cases of the rule in 1748 (and probably knew of it as early as 1729).

Consider a system of n linear equations for n unknowns, represented in matrix multiplication form as follows:

where the n by n matrix Failed to parse (Missing texvc executable; please see math/README to configure.): A

Then the theorem states that in this case the system has a unique solution, whose individual values for the unknowns are given by:

where Failed to parse (Missing texvc executable; please see math/README to configure.): A_i