Thus,A is obtained from A by letting row 1 of A be column 1 of 4',letting row 2 of A be column 2 of 47,and so on.For example, [14 if 4=斗 then A=25 36 Observe that ()T=4.Let B =[1 2];then BT= and 2 (BT=[12]=B As indicated by these two examples,for any matrix A,(4)T=A. Matrix Multiplication Given two matrices 4 and B,the matrix product of A and B(written AB)is defined if and only if Number of columns in 4=number of rows in B ) For the moment,assume that for some positive integer r,A has r columns and B has r rows.Then for some m and n,A is an m X r matrix and B is an r X n matrix. DEFINITION■ The matrix product C=AB of A and B is the m X n matrix C whose ijth element is determined as follows: ijth element of C scalar product of row i of A X column j of B(2) If Equation (1)is satisfied,then each row of A and each column of B will have the same number of elements.Also,if(1)is satisfied,then the scalar product in Equation(2) will be defined.The product matrix C=4B will have the same number of rows as 4 and the same number of columns as B. EXAMPLE 1 Matrix Multiplication Compute C=AB for [11 A= 「112] and 213 B=23 [12 Solution Because 4 is a 2 x 3 matrix and B is a 3 x 2 matrix,AB is defined,and C will be a 2 X 2 matrix.From Equation (2), [1 c11=[112] 2 1(1)+1(2)+2(1)=5 c12=[112]3 1(1)+1(3)+2(2)=8 2 [ c21=[213] 2=2(1)+1(2)+3(1)=7Thus, AT is obtained from A by letting row 1 of A be column 1 of AT , letting row 2 of A be column 2 of AT , and so on. For example, if A , then AT Observe that (AT ) T A. Let B [1 2]; then BT and (BT ) T [1 2] B As indicated by these two examples, for any matrix A, (AT ) T A. Matrix Multiplication Given two matrices A and B, the matrix product of A and B (written AB) is defined if and only if Number of columns in A number of rows in B (1) For the moment, assume that for some positive integer r, A has r columns and B has r rows. Then for some m and n, A is an m r matrix and B is an r n matrix. DEFINITION ■ The matrix product C AB of A and B is the m n matrix C whose ijth element is determined as follows: ijth element of C scalar product of row i of A column j of B ■ (2) If Equation (1) is satisfied, then each row of A and each column of B will have the same number of elements. Also, if (1) is satisfied, then the scalar product in Equation (2) will be defined. The product matrix C AB will have the same number of rows as A and the same number of columns as B. Compute C AB for A and B Solution Because A is a 2 3 matrix and B is a 3 2 matrix, AB is defined, and C will be a 2 2 matrix. From Equation (2), c11 [1 1 2] 1(1)  1(2)  2(1) 5 c12 [1 1 2] 1(1)  1(3)  2(2) 8 c21 [2 1 3] 2(1)  1(2)  3(1) 7 1 2 1 1 3 2 1 2 1 1 3 2 1 2 1 2 3 1 1 1 2 1 2 4 5 6 1 2 3 3 6 2 5 1 4 16 CHAPTER 2 Basic Linear Algebra EXAMPLE 1 Matrix Multiplication