574 CHAPTER 9 Mathematical Systems We can shorten the procedure as follows. A B 3 2 5 7 0 6 4 1 3(0) 2(4) 3(6) 2(1) 5(0) 7(4) 5(6) 7(1) 8 20 28 37 × = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = + + + + ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ In general, if A a b c d B e f g h and = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ then A B a b c d e f g h ae bg af bh ce dg cf dh × = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = + + + + ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ Let’s do one more multiplication of matrices. Example 6 Multiplying Matrices Determine A B × if A B 3 1 4 2 and 5 1 3 2 8 0 = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ Solution Matrix A contains two columns, and matrix B contains two rows. Thus, the matrices can be multiplied. Since matrix A contains two rows and matrix B contains three columns, the product matrix will contain two rows and three columns. × = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ = + − + + + − + + ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ A B 3 1 4 2 5 1 3 2 8 0 3(5) 1(2) 3( 1) 1(8) 3(3) 1(0) 4(5) 2(2) 4( 1) 2(8) 4(3) 2(0) 17 5 9 24 12 12 7 Now try Exercise 21 It should be noted that multiplication of matrices is not commutative; that is, A B B A, × ≠ × except in special instances.
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