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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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