Need help with homework - finding determinants

[Yuriman]

Problem:

Prove that (a-b)(b-c)(c-a) is the determinant of the matrix
|1 1 1 |
|a b c |
|a² b² c²|

I did matrices a few years back but have forgotten everything about them. I keep getting stuck. Please help!

 

[esun]

http://en.wikipedia.org/wiki/Determinant

 

[Yuriman]

Very helpful. I've read it, I'm making a mistake somewhere and it would be most helpful to me if I could see the problem worked in full so I can discover where my mistake is.

 

[yhelothar]

nm

 

[Fayd]

detA = bc^2 - b^2c - ac^2 + a^2c +ab^2 - a^2b.

multiply out your expression. what do you get?

 

[Fayd]

Convert to upper triangular form with elementary row operations and then multiply the diagonal \

i always hated that method...

 

[Yuriman]

Makes sense. I was coming at it from a different (the wrong?) direction.

I was trying to simplify bc^2 - b^2c - ac^2 + a^2c +ab^2 - a^2b into the other expression and was failing miserably. Is it possible to do so?

 

[Fayd]

it's certainly possible, but more trouble than it's worth. all you have to do to satisfy the prompt is prove the values are equal. if they can be represented the same way, regardless of what way they're presented in the prompt, then they're equal.

 

[yhelothar]

i always hated that method...

It's easier for some problems, especially 4x4 matricies.


This one looks like it could get pretty hairy with that though.

 

[Yuriman]

Alright, how about this one:
|x 0 c|
|-1 x b|
|0 -1 a| = 0

Thanks so much!

 

[Fayd]

It's easier for some problems, especially 4x4 matricies.


This one looks like it could get pretty hairy with that though.

not really. when calculating inverse matrices, you learn to love cofactors.

 

[esun]

Your problem has nothing to do with determinants, it's just basic algebra. If you can apply the formula in the determinant article then you should brush up on your algebra to understand how to demonstrate the equality of your expression and the answer.

 

[Brainonska511]

|1 1 1 |
|a b c |
|a² b² c²|

1*|b   c | - 1*|a  c | + 1*|a  b |
  |b2  c2|     |a2 c2|     |a2 b2|

Pick a row to work with, that becomes your coefficients (they go + - + -) and then the parts of the matrix that are not in the same column make up a new matrix to go with each coefficient.

Then reduce those determinants/matrices using ad - bc
eg: if the starting matrix was this

|a b|
|c d|

 

[Fayd]

Alright, how about this one:
|x 0 c|
|-1 x b|
|0 -1 a| = 0

Thanks so much!

i'm confused with what you're having trouble with...

 

[Yuriman]

Simplifying I get ax²+bx+c=0, but I suppose I'm confused as to what it's asking for. The problem simply says "solve" and I simplify to the generalized form for a quadratic.

 

[MovingTarget]

What course/level is this for? I know its linear algebra, but is this a graduate or undergraduate level course?

 

[Fayd]

What course/level is this for? I know its linear algebra, but is this a graduate or undergraduate level course?

gotta be undergrad.

 

[Fayd]

Simplifying I get ax²+bx+c=0, but I suppose I'm confused as to what it's asking for. The problem simply says "solve" and I simplify to the generalized form for a quadratic.

then i guess your answer is the quadratic equation. (-b +- sqrt b^2 etc etc.)

 

[Yuriman]

Thanks. I finished up my math a while back but my wife is taking it right now and she asked me for help. It's good to not look like the idiot I am. ^^