Cramer's rule gives a formula for the entries of a column vector x that satisfies Ax = b, where A is a n x n matrix with nonzero determinant and b is a column vector with n entries. A column vector in n x 1 matrix. Let A_j(b) denote the n x n matrix which has the same entries as A except for the jth column which is equal to b. Cramer's rule is that the jth entry of x must be equal to the det(A_j(b))/det A. The notation det(M) means the determinant of the matrix M.

Here is the proof of Cramer's rule. Let e_j be the length n column vector that has jth entry equal to one and all other entries equal to zero. The identity matrix is [e₁ e₂ ... e_n] = I. AI_j (x) = [Ae₁ ... Ax ... Ae_n] = A_j (b). Because the determinant of a product of matrices is equal to the product of their determinants, det(A)*det(I_j(x))=det(A_j(b)). Using cofactor expansion, it is easy to see that det(I_j(x)) = the jth entry of x. Therefore the jth entry of x = det(A_j(b))/det(A).

Cramer's rule is not very practical as an algorithm for solving the equation Ax = b because too many computations are involved. Generally, however, there is some margin of error in figuring b. So, it is often useful to know all the solutions for Ax = b as b varies over its margin of error. In this case, Cramer's rule is helpful for understanding the set of solutions.