Linear Algebra, Theory And Applications, 2012a

2.1. MATRICES 41
is it possible to multiply these matrices. According to the above discussion it should be a
2 × 3 matrix of the form
⎛
⎞
First column
{ }} {
( ) ⎛ Second column
1 2 1
⎝ 1 ⎞ { }} {
( ) ⎛ Third column
1 2 1
0 ⎠,
⎝ 2 ⎞ { }} {
( ) ⎛ 1 2 1
3 ⎠,
⎝ 0 ⎞
1 ⎠
⎜ 0 2 1
0 2 1
0 2 1 ⎟
⎝
−2
1
1 ⎠
You know how to multiply a matrix times a vector and so you do so to obtain each of the
three columns. Thus
( ) ⎛ 1 2 1 ⎝ 1 2 0
⎞
( )
0 3 1 ⎠ −1 9 3
= .
0 2 1
−2 7 3
−2 1 1
Here is another example.
Example 2.1.7 Multiply the following.
⎛
⎞
1 2 0 (
⎝ 0 3 1 ⎠ 1 2 1
0 2 1
−2 1 1
First check if it is possible. This is of the form (3 × 3) (2 × 3) . The inside numbers do not
match and so you can’t do this multiplication. This means that anything you write will be
absolute nonsense because it is impossible to multiply these matrices in this order. Aren’t
they the same two matrices considered in the previous example? Yes they are. It is just
that here they are in a different order. This shows something you must always remember
about matrix multiplication.
Order Matters!
Matrix multiplication is not commutative. This is very different than multiplication of
numbers!
2.1.1 The ij th Entry Of A Product
It is important to describe matrix multiplication in terms of entries of the matrices. What
is the ij th entry of AB? It would be the i th entry of the j th column of AB. Thus it would
be the i th entry of Ab j . Now
b j =
⎛
⎜
⎝
and from the above definition, the i th entry is
n∑
A ik B kj . (2.11)
k=1
B 1j
.
B nj
In terms of pictures of the matrix, you are doing
⎛
⎞ ⎛
⎞
A 11 A 12 ··· A 1n B 11 B 12 ··· B 1p
A 21 A 22 ··· A 2n
B 21 B 22 ··· B 2p
⎜ . .
. ⎟ ⎜ . . . ⎟
⎝ . .
. ⎠ ⎝ . . . ⎠
A m1 A m2 ··· A mn B n1 B n2 ··· B np
⎞
⎟
⎠
)