Problems 61–64 deal with the Vandermonde determinant

that will play an important role in Section 3.7.

The formulas in Problem 61 are the cases n = 2 and n = 3 of the general formula

Prove this as follows. Given x₁, x₂, and x₃, define the cubic polynomial P(y) to be

Because P(x₁) = P(x₂) = P(x₃) = 0 (why), the roots of P(y) are x₁, x₂, and x₃. It follows that

P(y) = k(y - x₁)(y - x₂)(y - x₃),

where k is the coefficient of y³ in P(y). Finally, observe that expansion of the 4 x 4 determinant in (26) along its last row gives k = V(x₁,x₂, x₃) and that V(x₁, x₂, x₃, x₄) = P(x₄).

Transcribed Image Text:

V(x1,X2,....Xn) = 1 X1 1 x2 기글 212 1 Xn ... 2 : .. -1 -1 -1