Chapter 4

Matrices
1.
Introduction
There are a certain number of terms that are used in matrix algebra with which you
need to become familiar.
Some of these terms are:
array,
dimension,
order,
element,
scalar,
vector,
identity matrix,
transpose,
inverse,
determinant,
reciprocal,
adjoint,
cofactor.
•
A matrix is a rectangular array of numbers, generally either one dimension or two
dimensions and is usually denoted by a letter in
bold
type,
e.g.,
A
=
3
1
5
4
2
9

b
=
[
]
1
3
4
1
2
c
=
5
2
1
Note that we have denoted a twodimensional matrix with a capital letter, e.g.
A
,
and a onedimensional matrix with a lowercase letter, e.g.
b.
•
One way of defining a matrix is by its dimensions.
Here the matrix,
A
, is said to
be
of order
2 x 3, as it has 2 rows and 3 columns.
Note we always state the
number of rows first, then the number of columns.
The matrix,
b
, is usually
referred to as a vector since
b
has only 1 row (its dimensions are 1 x 5 as it has
1 row and 5 columns).
Similarly,
c
is also a vector as it only has 1 column and
it is of order 3 x 1 as it has 3 rows and 1 column.
Example 1.
In recent years, the student enrolments in Econometrics in first , second and third
year have been as follows:
Year
First Year
Second Year
Third Year
1993
700
85
35
1994
750
90
30
1995
1050
120
45
1996
1200
150
60
We could represent this information in a matrix.
X
=
700
85
35
750
90
30
1050
120
45
1200
150
60
•
A matrix is also defined by its individual elements.
We denote the location (or
position) of an element in the matrix by specifying the row and column in
which the element is found.
For example, in the matrix
X
above, the number
750 is in the 2
nd
row and the 1
st
column.
We would denote this element as a
21
.
Note the row number is first and the column number is second.
In general
terms, the a
ij
element is in row i and column j.
Another example is that the
number 35 would be represented by a
13
as it is in the 1
st
row and the 3
rd
column.
When are two matrices equal?