572 CHAPTER 9 Mathematical Systems Multiplication of Matrices Multiplication of matrices is slightly more difficult than addition of matrices. Multiplication of matrices is possible only when the number of columns of the first matrix, A, is the same as the number of rows of the second matrix, B. We use the notation A 3 4 × to indicate that matrix A has three rows and four columns. Suppose matrix A is a 3 4 × matrix and matrix B is a 4 5 × matrix. Then × × A B 3 4 4 5 Same AB is a × 3 5 matrix This notation indicates that matrix A has four columns and matrix B has four rows. Therefore, we can multiply these two matrices. The product matrix, A B × or simply AB, will have the same number of rows as matrix A and the same number of columns as matrix B. Thus, the dimensions of the product matrix, AB , are 3 5. × b) We determined A3 in part (a). Now we find B2 . B A B 2 2 1 3 5 6 2( 1) 2(3) 2(5) 2(6) 2 6 10 12 3 2 3 12 9 15 2 6 10 12 3 ( 2) 12 6 9 10 15 12 5 6 19 3 = ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ − ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ − = − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ − ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ = − − − − − − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 7 Now try Exercise 17 Example 5 Can These Matrices Be Multiplied? Determine which of the following pairs of matrices can be multiplied to determine the product matrix AB. a) = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ A B 3 2 5 7 , 0 6 4 1 b) = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ A B 2 3 5 6 , 2 4 1 6 8 0 c) = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ A B 2 1 4 3 2 8 , 2 1 3 1 0 2 Solution a) × × A B 2 2 2 2 Same Because matrix A has two columns and matrix B has two rows, the two matrices can be multiplied. The product matrix, AB, is a 2 2 × matrix. Timely Tip Matrices can be added or subtracted only if they have the same dimensions. Matrices can be multiplied only if the number of columns in the first matrix is the same as the number of rows in the second matrix.
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