Uncovering the Pattern: Factoring and Vanishing Points in Determinants

[astonmartin]

Homework Statement

Alright so apparently there is some pattern in finding these determinants:

for the 2x2, the determinant of
|1 1|
|x y| is y - x

for 3x3, the determinant of

|1 1 1 |
|x y z |
|x^2, y^2, z^2| is xy^2 - yx^2 - xz^2 +zx^2 + yz^2 - zy^2

Apparently that can be factored (not sure how), and using the roots, there will be a pattern that you can observe for finding a determinant of a 4x4, 5x5, etc of the same form (1, x, x^2, x^3, etc.)

How do you factor the 3x3 determinant, and what is the pattern? Does the determinant vanish at some point?

 

[Dick]

Start with the 3x3 case. The pattern is that the determinant vanishes in the cases x=y, x=z and y=x. Do you see why? Each of those tells you a factor of the determinant.