Lin Algebra Row Reduction (on variables)

[Aerosion]

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]

Homework Equations

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]

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.

 

[Aerosion]

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).

 

[Aerosion]

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]

all you need is work it out and see...