Question

Show that  |(1, a, a²), (1, b, b²), (1, c, c²)| = (a - b) (b - c) (c - a).

Answer 1

Let

If b is put equal to a, two rows are exactly alike.

Therefore, Δ = 0 when b = a.

Hence, (a - b) is a factor of Δ [this follows from the factor theorem which states that for f(x), if f(a) = 0, then (x - a) is a factor of f(x)].

Similarly, (b - c) and (c - a) are factors.

Again, Δ is of third degree in a, b and c.

We already know the three linear factors are (a - b), (b - c) and (c - a). If there is another factor, it must be a mere number. Thus

= N (a - b) (b - c) (c - a), where N is a number

By equating coefficients of bc² on both sides, we get N = 1.

Therefore,

Δ = (a - b) (b - c) (c - a)

Alternative method:

Subtracting the second row from the first and then the third row from the second, we have

Now expanding along the first column, we have

Δ = (a - b) (b - c) [(b + c) - (a + b)] = (a - b) (b - c) (c - a)