A First Course in Linear Algebra, 2017a

2.1. Matrix Arithmetic 63
Definition 2.16 gives us a way to calculate each column of AB, asfollows.
⎡
⎤
First column
{ }} {
[ ] ⎡ Second column
⎤ { }} {
1 [ ] ⎡ Third column
⎤ { }} {
2 [ ] ⎡ ⎤
0
1 2 1
⎣ 1 2 1
0 ⎦,
⎣ 1 2 1
3 ⎦,
⎣ 1 ⎦
⎢ 0 2 1
0 2 1
0 2 1 ⎥
⎣
−2
1
1 ⎦
You know how to multiply a matrix times a vector, using Definition 2.13 for each of the three columns.
Thus
[ ] ⎡ ⎤
1 2 0 [ ]
1 2 1
⎣ 0 3 1 ⎦ −1 9 3
=
0 2 1
−2 7 3
−2 1 1
Since vectors are simply n × 1or1× m matrices, we can also multiply a vector by another vector.
Example 2.18: Vector Times Vector Multiplication
⎡ ⎤
1
Multiply if possible ⎣ 2 ⎦ [ 1 2 1 0 ] .
1
♠
Solution. In this case we are multiplying a matrix of size 3 × 1byamatrixofsize1× 4. The inside
numbers match so the product is defined. Note that the product will be a matrix of size 3 × 4. Using
Definition 2.16, we can compute this product as follows
⎡
⎡
⎣
1
2
1
⎤
First column Second column Third column Fourth column
⎤
{ }} {
⎦ [ 1 2 1 0 ] ⎡ ⎤ { ⎡ ⎤}} { { ⎡ ⎤}} { { ⎡ ⎤}} {
1
=
⎣ 2 ⎦ [ 1 ] 1
, ⎣ 2 ⎦ [ 2 ] 1
, ⎣ 2 ⎦ [ 1 ] 1
, ⎣ 2 ⎦ [ 0 ] ⎢
⎥
⎣ 1 1 1 1 ⎦
You can use Definition 2.13 to verify that this product is
⎡
1 2 1
⎤
0
⎣ 2 4 2 0 ⎦
1 2 1 0
♠
Example 2.19: A Multiplication Which is Not Defined
Find BA if possible.
⎡
B = ⎣
1 2 0
0 3 1
−2 1 1
⎤
⎦,A =
[
1 2 1
0 2 1
]