ECON211 Chapter 5: Cramer's Rule
27 Jul 2016
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Consider the system of linear equations ax = b where a is n by n, and x and b are n by. If the determinant of a is not zero we have seen that the solution for x is. 1 det(a) c11 c12 c1n c21 c22 c2n cn1 cn2 cnn. A = [aij] and cij is the cofactor of aij. Thus the ith row of the solution vector x is xi = 1 det(a) (c1ib1 + c2ib2 + + cnibn) . (1) Cramer"s rule says that xi = det(a )/ det(a) where a is a with the ith column replaced by b, that is, xi = 1 det(a) det a11 a21 an1 b1 b2 bn a1n a2n ann. Using the ith column to calculate det(a ) we obtain xi = 1 det(a) (b1c1i + b2c2i + + bncni) , which is the same as (1).
