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Chapter 4 -- Matrices
The theory of matrices, discovered by Arthur Cayley in the 19th century and advanced by
others, was initially regarded as a mathematical curiosity without much practical application.
However, since the Second World War, developments in matrix algebra have allowed it to
become a powerful tool for summarizing and analysing systems of equations.
In this
introduction to the topic, we hope to show some of its utility, however much of its power
must be left to higher level courses.
1.
Introduction
A matrix is a rectangular array of numbers, generally either one dimension or two dimensions,
e.g.,
⎡
5
⎤
A
=
⎡
3 -1 5
⎤
b
= [1 3 4 1
2]
c
=
⎢
2
⎥
⎣
4 2 9
⎦
⎣
1
⎦
We can represent student enrolments by a matrix
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, say,
X
=
700
85
35
750
90
30
1050 120
45
1200 150
60
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
A matrix is generally denoted by a letter in
bold
type.
We will follow the convention which
denotes a two dimensional matrix by a capital letter (e.g.,
A
) and a one with only a single row
or column (called a vector) by a lower case letter (e.g.,
b
,
c
).
A matrix is defined not only by its elements, but by its dimensions, i. e., the number of rows
and columns it possesses.
The matrix
A
above is said to be of
order
2 x 3, as it has two rows
and three columns.
(Sometimes the term
dimension
is used interchangeably with the term
order
.
Thus
A
could be called a two dimensional matrix, and it would be said to have
dimension 2x3.)
Similarly
b
is said to be a matrix of
order
1 x 5 (or equivalently, a row
vector of
order
5).
The location within a matrix of an element, a
ij
, is denoted by its row (i) and column (j)
subscripts.
The first subscript is that of the row, and the second subscript is that of the
column.
Thus for the matrix defined by
A
above
the element a
21
is the value 4, as it is in the second row, first column
the element a
13
is the value 5 as it is in the first row, second column.