Advanced Calculus fi..

Chapter 1 Vectors and Matrices 27Conversely, if det A # 0, then A has an inverse. For if det A # 0, then thesimultaneous linear equationsall.rl + . . .+ alnxn = YIUnlX1 + ... + UnnXn = Yncan be solved for xl, ..., xn by Cramer's Rule (Section 1.5). For example,(1.62)where D = det A. Upon expanding the first determinant on the right, we obtain anexpression of the formwith appropriate constants bll, . . . , bin. In general,Now our given equations (1.62) are equivalent to the matrix equationand the solution (1.63) is given byAx = y,x = By.The fact that this is a solution is expressed by the relationABy=y or ABy= ly.This relation holds for all y. Hence by Rule 20 of Section 1.8 we must have A B = I,so that B is an inverse of A.The reasoning just given also provides a constructive way of finding Apl. Onesimply forms the equations (1.62) and solves for XI, ..., x,. The solution can bewritten as x = By, where B = A-'.EXAMPLE 1A = [: :]. The simultaneous equations areWe solve by elimination, and find2x1 + 5~2 = y1, XI + 3x2 = Y?.Therefore A-' = [ i]. We check by verifying that AA-' = I .