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.
introduction to the topic, we hope to show some of its utility, however much of its power
must be left to higher level courses.
A matrix is a rectangular array of numbers, generally either one dimension or two dimensions,
3 -1 5
= [1 3 4 1
4 2 9
We can represent student enrolments by a matrix
We could represent this information in a matrix, say,
A matrix is generally denoted by a letter in
We will follow the convention which
denotes a two dimensional matrix by a capital letter (e.g.,
) and a one with only a single row
or column (called a vector) by a lower case letter (e.g.,
A matrix is defined not only by its elements, but by its dimensions, i. e., the number of rows
and columns it possesses.
above is said to be of
2 x 3, as it has two rows
and three columns.
(Sometimes the term
is used interchangeably with the term
could be called a two dimensional matrix, and it would be said to have
is said to be a matrix of
1 x 5 (or equivalently, a row
The location within a matrix of an element, a
, is denoted by its row (i) and column (j)
The first subscript is that of the row, and the second subscript is that of the
Thus for the matrix defined by
the element a
is the value 4, as it is in the second row, first column
the element a
is the value 5 as it is in the first row, second column.