SECTION 3.7 Discriminant 175
V = det
1 x 1 x^21 ··· xnn−^1
1 x 2 x^22 ··· xn 2 −^1
... ...1 xn x^2 n ··· xnn−^1
which makes frequent appearances throughout mathematics. Its deter-
minant is given in the next theorem.
Theorem 3. detV =
∏
i<j(xj−xi).Proof. We argue by induction onn. Setting ∆ = detV, we start by
subtracting row 1 from rows 2, 3 ,...,n, which quickly produces
∆ = det
x 2 −x 1 x^22 −x^21 ··· xn 2 −^1 −xn 1 −^1
x 3 −x 1 x^23 −x^21 ··· xn 3 −^1 −xn 1 −^1
... ... ... ...
xn−x 1 x^2 n−x^21 ··· xnn−^1 −xn 1 −^1
.
Next, in each row we factor out the common factor of xi−x 1 , i =
2 , 4 ,...,n, which leads to
∆ = (x 2 −x 1 )(x 3 −x 1 )···(xn−x 1 )×det
1 x 2 +x 1 x^22 +x 2 x 1 +x^21 ··· xn 2 −^2 +xn 2 −^3 x 1 +···+xn 1 −^2
1 x 3 +x 1 x^23 +x 2 x 1 +x^21 ··· xn 3 −^2 +xn 3 −^3 x 1 +···+xn 1 −^2
... ... ... ... ...
1 xn+x 1 x^2 n+xnx 1 +x^21 ··· xnn−^2 +xnn−^3 x 1 +···+xn 1 −^2
.
Next, if we subtractx 1 times columnn−2 from column n−1, then
subtractx 1 times column n−3 from column n−2, and so on, we’ll
eventually reach
∆ = (x 2 −x 1 )(x 3 −x 1 )···(xn−x 1 ) ×det
1 x 2 x^22 ··· xn 2 −^2
1 x 3 x^23 ··· xn 3 −^2
... ... ... ... ...
1 xn x^2 n ··· xnn−^2
= (x 2 −x 1 )(x 3 −x 1 )···(xn−x 1 )×∏
j>i≥ 2(xj−xi) =∏
j>i(xj−xi),