Page 18 - A Course in Linear Algebra with Applications

2 Chapter One: Matrix Algebra

Explicit examples of matrices are

/ 4 3 \ , / 0 2.4 6 \
[l 2) a n d {^=2 3/5 - l j -

Example 1.1.1
Write down the extended form of the matrix (-l)*j + 1)3,2 •
(
l
The (i,j) entry of the matrix is (—l)j + i where i — 1,
2, 3, and j — 1, 2. So the matrix is

(1 "0-

It is necessary to decide when two matrices A and B are
to be regarded as equal; in symbols A = B. Let us agree this
will mean that the matrices A and B have the same numbers
j
of rows and columns, and that, for all i and , the (i,j) entry
of A equals the (i,j) entry of B. In short, two matrices are
equal if they look exactly alike.
As has already been mentioned, matrices arise when one
has to deal with linear equations. We shall now explain how
this comes about. Suppose we have a set of m linear equations
in n unknowns xi, X2, •••, x n. These may be written in the
form

{ anxi + CL12X2 + + • • • • • + • + a 2nXn a\ nx n = = £>2 bi
a 22X2
+
CL21X1
o m iXi + a m2x 2 + • • • + a

Here the a^ and bi are to be regarded as given numbers. The
problem is to solve the system, that is, to find all n-tuples
of numbers xi, x 2, ..., x n that satisfy every equation of the