Advanced Calculus fi..

22 Advanced Calculus, Fifth EditionFigure 1.14Product of two matricesWe observe that (ai I, . . . , a;,) is the ith row vector of A and that col (bl, , . . . . b,)is the jth column vector of B. Hence to form the product A B = C = (c,,), we obtaineach c,, by multiplying corresponding entries of the ith row of A and the jth columnof B and adding. The process is suggested in Fig. 1.14.We remark that the product A B is defined only when the number of columns ofA equals the number of rows of B; that is, when A is m x p and B is p x n, AB isdefined and is m x n. Also, when A B is defined, BA need not be defined, and evenwhen it is, A B is generally not equal to BA; that is, there is no cornrnutative law formultiplication.EXAMPLE 1Herem = 2, p = 2,n = 3.EXAMPLE 2Herem = 3,p = 3,n = 1.The second example illustrates the important case of the product Av, where A isan m x n matrix and v is an n x 1 column vector. The product Av is again a columnvector u, m x 1.In the general product A B = C, as defined above, we note that the jth columnvector of C is formed from A and the jth column vector of B, for the jth columnvector of C is