Our experience so far suggests to find for systems of linear equations of order n > 3 an expression dn, which performs the same task as d2 and d3 for systems of two and three equations. Let us take a close look at d2 and d3.
In the sums d2 and d3, every term is the product of two and of three of the aik, respectively. Moreover, you observe immmediately that there appears in each term in the sums from each row of the corresponding matrix exactly one factor. Indeed, in d2, each term contains a single a1k and a single a2l, in d3, a single a1k, a single a2l and a single a3m. In these formulae, we have chosen for each term in the sums an arrangement of the first subscripts in the natural sequence. Moreover, note that in each term all the second subscripts of the factors differ. Consider now the sequence of the second subscripts. In the two terms of d2, the second subscripts are
1,2 and 2,1,
in the six terms of d3, they are
[1,2,3]; [2,3,1]; [3,1,2]; [1,3,2]; [2,1,3]; [3,2,1].
You see that the second subscripts appear in all their different permutations.
Finally, consider the signs of the individual terms in the sums. Since they only differ by the permutation of the second subscripts, you will suspect that the signs are related to the permutations. This suspicion is confirmed by d2 as well as d3. In fact, in the terms with positive signs, the second subscripts form even, in those with negative sign, odd permutations. Of course, it is assumed here that the first subscripts of each term are in the natural sequence.
Having sufficiently analyzed the structures of d2 and d3, we can now form for every natural number n an analogous expression. For this purpose, we start from a matrix
with n rows and n columns. Each term in dn must have, apart from its sign, the form
where [p1, p2, ··· , pn] is a permutation of the numbers 1, 2, ··· n. Like in d2 and d3, we select as the sign of these terms the symbol c (p1, p2, ··· , pn) of the permutation [p1, p2, ··· , pn], so that
This
summation is over all n! permutations P of the
numbers 1, 2, ··· , n.
Following Leibniz, who also was solving linear equations,
you call dn the determinant of the matrix A
and employ the notation
In order to indicate that |A| has n rows and n columns, you talk of a determinant of order n.
In Section 12, which deals with the general theory of such systems, we will see that dn actually has for the solution of systems of simultaneous linear equations with n unknowns the same role as d2 and d3 for systems of two and three equations. Note also that the importance of determinants is not restricted to this function.
Exercises:
6. What is the value of a determinant in which all elements above ( below) the main diagonal vanish? (Find the answer using Definition (6.1) without using the result presented in Section 11.)
7. What can be the value of a determinant in which every line and every column has a single 1 and otherwise only zeroes (the socalled permutation matrix).
8. Let the elements of a determinant |A| be differentiable functions aik(x). Find d|A)|dx.