8.2. MULTIPLICATION OF MATRICES 133
Definition 8.2.1 An m×n matrix is a rectangular array of numbers which has m rows andn columns. We write this as
A =
A11 A12 · · · A1n
A21 A22 · · · A2n...
......
Am1 An2 · · · Amn
Thus the entry in the ith row and the jth column is denoted as Ai j. As suggested above,
A x=
A11 A12 · · · A1n
A21 A22 · · · A2n...
......
Am1 An2 · · · Amn
x1
x2...
xn
= x1
A11
A21...
Am1
+ x2
A12
A22...
An2
+ · · ·+ xn
A1n
A2n...
Amn
Note that Ax is in Fm and the ith entry of this vector Ax is
Ai1x1 +Ai2x2 + · · ·+Ainxn =n
∑j=1
Ai jx j.
In other words, the ith entry of Ax is the dot product of the ith row of A with the vector x.Symbolically,
(Ax)i = ∑j
Ai jx j (8.9)
We like to write x to denote an n× 1 matrix which is often called a vector. Then xT willdenote a 1×n matrix or row vector.
Example 8.2.2 (1 2 11 0 2
) x1
x2
x3
= x1
(11
)+ x2
(20
)+ x3
(12
)=
(x1 +2x2 + x3
x1 +2x3
)Note that if A is m×n then Ax is an m×1 matrix provided x is n×1. Thus A makes a
vector in Fn into a vector in Fm.
Example 8.2.3 Show the following: 1 −1 23 2 12 3 −3
1
23
=
510−1