# Lin Algebra Row Reduction (on variables)

## Homework Statement

Use row reduction to show that:

[1 1 1]
[a b c] = (b-a)(c-a)(c-b)
[a^2 b^2 c^2]

## The Attempt at a Solution

So I multiplied -a by row 1 and added it to row 2. I then multiplied -a^2 by row 1 and added it to row 3, which gave me:

[1 1 1]
[0 b-a c-a]
[0 b^2-a^2 c^2-a^2]

Now, at this point, I was already quite confused, as I don't know how I did that (I was following the book). Where did the -a and -a^2 come from? Thin air? How was I to know to multiply a (as opposed to b or c)?

Anyway, after that, I did cofactor expansion and expanded the b^2-a^2 and the c^2-a^2 to get (b-a)(b+a) and (c-a)(c+a). This is pretty much where I got up to. I think I'm close, but I don't know what to do next. Suggestions? Advice? Hmm?

mjsd
Homework Helper
I think they are asking you to find the determinant of the matrix using row reduction. get the matrix in upper triangular form then product of diagonal entries is the determinant.

Well, I'm not sure how to put it in upper triangular form, as I'm not sure how to get these numbers to equal zero (unless I multiply -a to the first row, and I'm not sure if I can do that, since I don't know if you can multiply in row reduction).

Wait...R2 - aR1 and R3 - a^2R1, right? That would explain the first column...and how I got the second matrix...

Then do the cofactor expansion and take the determinant, which gets an equation that can factor out to (b-a)(c-a)(c-b)...is that right?

mjsd
Homework Helper
all you need is work it out and see....