[ c22=[213] 3 =2(1)+1(3)+3(2)=11 2 C=AB 5 81 > 11 EXAMPLE 2 Column Vector Times Row Vector Find AB for A= [ and B=[12] Solution Because A has one column and B has one row,C=AB will exist.From Equation(2),we know that C is a 2 X 2 matrix with c11=3(1)=3 c21=4(1)=4 c12=3(2)=6 C22=4(2)=8 Thus, c- EXAMPLE 3 Row Vector Times Column Vector Compute D B4 for the A and B of Example 2. Solution In this case,D will be a 1 X 1 matrix (or a scalar).From Equation(2), d=1 T31 24=13)+244=11 Thus,D=[11].In this example,matrix multiplication is equivalent to scalar multiplica- tion of a row and column vector. Recall that if you multiply two real numbers a and b,then ab =ba.This is called the commutative property of multiplication.Examples 2 and 3 show that for matrix multipli- cation,it may be that 4B BA.Matrix multiplication is not necessarily commutative.(In some cases,however,AB =BA will hold.) EXAMPLE 4 Undefined Matrix Product Show that 4B is undefined if 4-6 [1 and B=01 12 Solution This follows because 4 has two columns and B has three rows.Thus,Equation(1)is not satisfied.c22 [2 1 3] 2(1)  1(3)  3(2) 11 C AB Find AB for A and B [1 2] Solution Because A has one column and B has one row, C AB will exist. From Equation (2), we know that C is a 2 2 matrix with c11 3(1) 3 c21 4(1) 4 c12 3(2) 6 c22 4(2) 8 Thus, C Compute D BA for the A and B of Example 2. Solution In this case, D will be a 1 1 matrix (or a scalar). From Equation (2), d11 [1 2] 1(3)  2(4) 11 Thus, D [11]. In this example, matrix multiplication is equivalent to scalar multiplica￾tion of a row and column vector. Recall that if you multiply two real numbers a and b, then ab ba. This is called the commutative property of multiplication. Examples 2 and 3 show that for matrix multipli￾cation, it may be that AB BA. Matrix multiplication is not necessarily commutative. (In some cases, however, AB BA will hold.) Show that AB is undefined if A and B Solution This follows because A has two columns and B has three rows. Thus, Equation (1) is not satisfied. 1 1 2 1 0 1 2 4 1 3 3 4 6 8 3 4 3 4 8 11 5 7 1 3 2 2.1 Matrices and Vectors 17 EXAMPLE 3 Row Vector Times Column Vector EXAMPLE 4 Undefined Matrix Product EXAMPLE 2 Column Vector Times Row Vector