6.5 Properties and Applications of Matrices
A matrix is a rectangular array of numbers.
The anatomy of a matrix
col
1
col
col
2
3

3
 5
1
row 1 

 =

1
3 
row 2   2
each element
has a name:
 a11

a
 21
a12
a 22
a13 

a 23 
row dimension: 2
column dimension: 3
dimension of matrix: 2  3 ("two by three")
a square matrix has the same number of rows as columns
Operations on matrices
1. Addition and subtraction: (element-by-element)
1

2

3
1
 5  0
+
3   3
1
0
 2  1
 =
1   __
2
__
__ 

__ 
If dimensions are different, addition/subtraction undefined
Two matrices are equal if:
 they have the same dimensions
 all corresponding elements are equal
6.5-1
2. Scalar multiplication
Multiply a matrix by a number (element-by-element):
1
3
2

3
1
 5
 3
 = 
 __
3 

9
__
__ 

__ 
3. Matrix multiplication (row-by-column inner products):
column 1
element (1,1)


row 1
3
0
1

  1 3  5    1  1 0  =   12

2

 
1
3  



2

0
0


element (1,2)
column 2

row 1

3
0
1

  1 3  5    1  1 0  =   12 0

2




1
3 



2

0
0

Important: for above product we have:
(2 x 3 ) matrix  ( 3 x 3) matrix
 boxed (inside) numbers have to be the same, or . . .
 product is undefined (inner product does not compute)
Note: ( 2 x 3)  ( 3 x 3 )
= ()
 boxed (outside) numbers give us 2 x 3
 = dimension of the product
6.5-2
Application of matrix multiplication
The scenario:
 two factories, one in Austin, one in San Antonio
 each manufactures two kinds of skis: trick and slalom
 two manufacturing steps: fabricate and finish
 per-hour labor costs differ by step and location
 here are the numbers, arranged in two matrices:
Time matrix
Hourly labor costs matrix
fabricate
finish
trick
6
1.5
fabricate
slalom
4
1
finish
Austin
San Antonio
10
12
8
10
What does the product of above (2 x 2) matrices represent?
Consider product element a11 = (6)(10) + (1.5)(8) = 72
Does this inner product have a meaningful interpretation?
Here’s the whole product matrix:
Austin
San Antonio
trick
72
87
slalom
48
58
How would you describe what this matrix represents?
What good is all this?
6.5-3