CRAMER'S RULE

in linear and multilinear algebra, procedure for solving systems of simultaneous linear equations by means of determinants (see also determinant; linear equation). If is a system of n simultaneous linear equations in n unknowns, then a solution of this system is in which det A is the determinant of the matrix A (in which the elements of each row are the coefficients aij of one of the equations) and the matrix Bi is formed by replacing the ith column of A by the column of constants b1, . . . , bn. If det A equals zero, the system has no unique solution; that is, there is no set x1, . . . , xn that satisfies all of the equations.