Page 91 - A Course in Linear Algebra with Applications

3.2: Basic Properties of Determinants 75

is n + 1.
First note the obvious equalities D\ — 2 and D2 3. Let
n > 3; then, expanding by row 1, we obtain

1 1 0 0 • • 0 0 0
0 2 1 0 • • 0 0 0
0 1 2 1 • • 0 0 0
— 2D n-\ —
D n
0 0 0 0 • • 1 2 1
0 0 0 0 • • 0 1 2

Expanding the determinant on the right by column 1, we find
it to be D n _2- Thus

= 2D n _! — D n-2-
D n
This is a recurrence relation which can be used to solve for
successive values of D n. Thus D3 = 4 , D4 = 5, D5 = 6 , etc.
In general D n = n + 1. (A systematic method for solving
recurrence relations of this sort will be given in 8.2.)

The next example is concerned with an important type
of determinant called a Vandermonde determinant; these de-
terminants occur frequently in applications.

Example 3.2.4
Establish the identity

1 1
Xi X2
Xn X„ n , Xj, Xj),
1,3
n - 1 n - 1
X JUn X

where the expression on the right is the product of all the
factors Xi — Xj with i < j and i,j = l,2,...,n.