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Stitz-Zeager_College_Algebra_e-book

# 7 we use row operations to transform our 3 3 matrix a

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Unformatted text preview: mn: Suppose A = [aij ]m×n and B = [bij ]n×r . Let Ri denote the ith row of A and let Cj denote the j th column of B . The product of Ri and Cj , denoted Ri · Cj is the real number deﬁned by Ri · Cj = ai1 b1j + ai2 b2j + . . . ain bnj Note that in order to multiply a row by a column, the number of entries in the row must match the number of entries in the column. We are now in the position to deﬁne matrix multiplication. Definition 8.10. Matrix Multiplication: Suppose A = [aij ]m×n and B = [bij ]n×r . Let Ri denote the ith row of A and let Cj denote the j th column of B . The product of A and B , denoted AB , is the matrix deﬁned by AB = [Ri · Cj ]m×r that is AB = R1 · C 1 R2 · C 1 . . . R1 · C 2 R2 · C 2 . . . ... ... R1 · Cr R2 · Cr . . . Rm · C 1 Rm · C 2 . . . Rm · Cr There are a number of subtleties in Deﬁnition 8.10 which warrant closer inspection. First and foremost, Deﬁnition 8.10 tells us that the ij -entry of a matrix product AB is the ith row of A times the j th column of B . In order for this to be d...
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