Page 81 - A Course in Linear Algebra with Applications

3.1: The Definition of a Determinant 65

a
a
ana220-33 + a 1 2 2 3 3 1 + ai3G21<332
— 012021033 — 013022^31 — ^Iia23tl32
We could in a similar fashion write down the general 4 x 4 de-
terminant as a sum of 4! =24 terms, 12 with a positive sign
and 12 with a negative sign. Of course, it is clear that the
definition does not provide a convenient means of comput-
ing determinants with large numbers of rows and columns;
we shall shortly see that much more efficient procedures are
available.
Example 3.1.4
What term in the expansion of the 8x8 determinant det((ajj))
corresponds to the permutation 8, 3, 2, 6, 5, 1, 4, 7 ?

We saw in Example 3.1.1 that this permutation is odd,
so its sign is —1; hence the term sought is

a
~ Ctl8a23 32«46a55«6l074^87-
Minors and cofactors
In the theory of determinants certain subdeterminants
called minors prove to be a useful tool. Let A = (a^) be an
n x n matrix. The (i, j) minor M i;- of A is defined to be the
determinant of the submatrix of A that remains when row i
and column j of A are deleted.
The (i, j) cofactor Aij of A is simply the minor with an
appropriate sign:

Ay = ( - l r ^ M y .
For example, if
( a n «12 a23 J ,
a 13\
a 2\
a 22
«31 «32 G 3 3 /