Page 19 - A Course in Linear Algebra with Applications

1.1: Matrices 3

system, or to show that no such numbers exist. Solving a set
of linear equations is in many ways the most basic problem of
linear algebra.
The reader will probably have noticed that there is a ma-
trix involved in the above linear system, namely the coefficient
matrix

= =
•"• y&ij )m,n-
In fact there is a second matrix present; it is obtained by using
the numbers bi, b2, ••., b mto add a new column, the (n + l)th,
to the coefficient matrix A. This results in an m x (n + 1)
matrix called the augmented matrix of the linear system. The
problem of solving linear systems will be taken up in earnest
in Chapter Two, where it will emerge that the coefficient and
augmented matrices play a critical role. At this point we
merely wish to point out that here is a natural problem in
which matrices are involved in an essential way.

Example 1.1.2

The coefficient and augmented matrices of the pair of linear
equations
2xi —3x 2 +5a; 3 = 1
^ -x x + x 2 - x 3 = 4

are respectively

2 - 3 5\ , f 2 -3 5 1
and
- 1 1 - 1 7 V 1 1 - 1 4
-

Some special matrices

Certain special types of matrices that occur frequently
will now be recorded.
(i) A 1 x n matrix, or n — row vector, A has a single row

A = (an a 12 ... a ln).