Page 83 - A Course in Linear Algebra with Applications

3.1: The Definition of a Determinant 67

where the summation is now over all permutations 12, • • •, i n
a
of the integers 2,..., n. But the coefficient of n in this last
expression is just Mu = An. Hence the coefficient of an is
the same on both sides of the equation in (i).
We can deduce the corresponding statement for general i
and k by means of the following device. The idea is to move
dik to the (1, 1) position of the matrix in such a way that it
will still have the same minor M^. To do this we interchange
row % of A successively with rows i — l,i — 2,...,l, after which
dik will be in the (1, k) position. Then we interchange column
k with the columns k — l,k — 2,...,l successively, until ^ is in
a
the (1,1) position. If we keep track of the determinants that
arise during this process, we find that in the final determinant
the minor of a ik is still M^. So by the result of the first
paragraph, the coefficient of a^ in the new determinant is
M ik.
However each row and column interchange changes the
sign of the determinant. For the effect of such an interchange
is to switch two entries in every permutation, and, as was
pointed out during the proof of 3.1.3, this changes a permu-
tation from even to odd, or from odd to even. Thus the sign
of each permutation is changed by —1. The total number of
interchanges that have been applied is (i — 1) + (k — 1) =
i + k — 2. The sign of the determinant is therefore changed by
(
(-l) i + f c ~ 2 = -l) i + f c . It follows that the coefficient of a^ in
det(A) is (—l) l+k Mik, which is just the definition of An-.

(It is a good idea for the reader to write out explicitly
the row and column interchanges in the case n — 3 and i = 2,
k = 3, and to verify the statement about the minor M23).
The theorem provides a practical method of computing
3 x 3 determinants; for determinants of larger size there are
more efficient methods, as we shall see.