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Unformatted text preview: Chapter 2.5: Matrix Multiplication In the last section, we looked at the operations of transposition, scalar multiplication, matrix addition and matrix subtraction. Today, we will look at the multiplication of two matrices. This operation is perhaps not as straightforward as addition and subtraction. We will first look at the simplest type of matrix multiplication, multiplying a row matrix and a column matrix. As with addition and subtraction, there is a size requirement on this type of multiplication. Both the row matrix and the column matrix must have the same number of entries. That is, if the row matrix is a 1 n matrix, then the column matrix must be a n 1 matrix. Let A = bracketleftbig a 1 a 2 a 3 . . . a n bracketrightbig . Let B = b 1 b 2 b 3 . . . b n . We multiply the matrices A and B by multiplying together the corresponding entries, then adding up these products. That is, AB = a 1 b 1 + a 2 b 2 + + a n b n . The order of these two matrices is very important. We may multiply rows by columns, but never columns by rows. The product of a column multiplied by a row is undefined . Example: Let A = bracketleftbig 5 2 3 1 10 bracketrightbig . Let B = 3 1 8 10 . Find the product AB . We will multiply the corresponding entries of A and B , and add these results together. AB = (5)(3) + ( 2)( 1) + (3)(0) + ( 1)(8) + (10)( 10) 1 = 15 + 2 + 0 8 100 = 91 For any two matrices A and B , we will define the product AB in terms of rows of A multiplied by columns of B . In order for the product AB to be defined, we will need the number of entries in each row of A to be equal to the number of entries in each column of B . A quick way to decide whether or not AB is defined is by writing the size of A and the size of B next to each other. If the inner two numbers match, then the product is defined....
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