A First Course in Linear Algebra, 2017a

2.1. Matrix Arithmetic 65
The j th column of AB is of the form
⎡
⎤⎡
a 11 a 12 ··· a 1n
a 21 a 22 ··· a 2n
⎢
⎣
.
. . ..
⎥⎢
. ⎦⎣
a m1 a m2 ··· a mn
⎤
b 1 j
b 2 j
⎥
. ⎦
b nj
which is an m × 1 column vector. It is calculated by
⎡ ⎤ ⎡ ⎤ ⎡
a 11 a 12
a 21
b 1 j ⎢ ⎥
⎣ . ⎦ + b a 22
2 j ⎢ ⎥
⎣ . ⎦ + ···+ b nj⎢
⎣
a m1 a m2
⎤
a 1n
a 2n
⎥
. ⎦
a mn
Therefore, the ij th entry is the entry in row i of this vector. This is computed by
a i1 b 1 j + a i2 b 2 j + ···+ a in b nj =
n
∑ a ik b kj
k=1
The following is the formal definition for the ij th entry of a product of matrices.
Definition 2.21: The ij th Entry of a Product
Let A = [ a ij
]
be an m × n matrix and let B =
[
bij
]
be an n × p matrix. Then AB is an m × p matrix
and the (i, j)-entry of AB is defined as
Another way to write this is
(AB) ij =
(AB) ij = [ ]
a i1 a i2 ··· a in ⎢
⎣
⎡
n
∑ a ik b kj
k=1
⎤
b 1 j
b 2 j
.
.
⎥
b nj
⎦ = a i1b 1 j + a i2 b 2 j + ···+ a in b nj
In other words, to find the (i, j)-entry of the product AB, or(AB) ij , you multiply the i th row of A, on
the left by the j th column of B. To express AB in terms of its entries, we write AB = [ ]
(AB) ij .
Consider the following example.
Example 2.22: The Entries of a Product
Compute AB if possible. If it is, find the (3,2)-entry of AB using Definition 2.21.
⎡ ⎤
1 2 [ ]
A = ⎣
2 3 1
3 1 ⎦,B =
7 6 2
2 6