Page 73 - A Course in Linear Algebra with Applications

Chapter Three

DETERMINANTS

Associated with every square matrix is a scalar called the
determinant. Perhaps the most striking property of the de-
terminant of a matrix is the fact that it tells us if the matrix
is invertible. On the other hand, there is obviously a limit
to the amount of information about a matrix which can be
carried by a single scalar, and this is probably why determi-
nants are considered less important today than, say, a hundred
years ago. Nevertheless, associated with an arbitrary square
matrix is an important polynomial, the characteristic poly-
nomial, which is a determinant. As we shall see in Chapter
Eight, this polynomial carries a vast amount of information
about the matrix.

3.1 Permutations and the Definition of a Determinant
Let A = (aij) be an n x n matrix over some field of scalars
(which the reader should feel free to assume is either R or C).
Our first task is to show how to define the determinant of A,
which will be written either

det(A)

or else in the extended form
an ai2
Q21 «22 &2n

O-nl a n2
For n = l and 2 the definition is simple enough:

kill = an and a u a i2 ^ 1 1 ^ 2 2 — ^ 1 2 0 2 1 -
«2i a-ii

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