Proving a determinant

[jman2807]

The question asks to reduce the the matrix down to triangular form and then show that

 _                   _
|                      |
| 1     1     1      |
|                      |
| a     b     c      | = (b-a)(c-a)(c-b)
|                      |
|a^2  b^2  c^2 |
|_                   _|

I don't really know how to begin to reduce this to triangular form.

Thanks in advance.[/code]

 

[Deleted member 4993]

The question asks to reduce the the matrix down to triangular form and then show that

 _                   _
|                      |
| 1     1     1      |
|                      |
| a     b     c      | = (b-a)(c-a)(c-b)
|                      |
|a^2  b^2  c^2 |
|_                   _|

I don't really know how to begin to reduce this to triangular form.

Thanks in advance.

What does triangular form mean?

Which triangular form - upper or lower?

What type of reduction techniques you have been taught?

 

[jman2807]

upper triangular form...

* * *
0 * *
0 0 *

All of the previous reduction techniques have had numbers involved and I don't really know how to begin to reduce this one.

 

[Deleted member 4993]

The question asks to reduce the the matrix down to triangular form and then show that

 _                   _
|                      |
| 1     1     1      |
|                      |
| a     b     c      | = (b-a)(c-a)(c-b)
|                      |
|a^2  b^2  c^2 |
|_                   _|

Exactly as if you had numbers.

start by multiplying row 1 by 'a' and subtract from row 2

 _                           _
|                              |
| 1         1         1        |
|                              |
| 0        b-a       c-a       | 
|                              |
|a^2       b^2       c^2       |
|_                           _ |

Then multiply row 1 by a^2 and subtract from row 3 .... and continue

 

[jman2807]

Well once i have done those steps i cant find another move without making the thing extremely complicated

 _                                       _
|                                         |
|     1              1              1     |
|                                         | 
|     0             (b-a)        (c-a)    |
|                                         |
|    0          (b^2-a^2)        (c^2-a^2)|
|_                                       _|

I thought of splitting up b^2-a^2 into (b-a)(b+a) then multiplying row 2 times (b+a) and subtracting from row three but then the last element in row three would be (c^2-a^2) - (c-a)(b+a)... Taking the determinant here yields nothing productive. Stuck again.. Thanks very much for the help so far though.[/code]

 

[Deleted member 4993]

Well once i have done those steps i cant find another move without making the thing extremely complicated

 _                                       _
|                                         |
|     1              1              1     |
|                                         | 
|     0             (b-a)        (c-a)    |
|                                         |
|    0          (b^2-a^2)        (c^2-a^2)|
|_                                       _|

I thought of splitting up b^2-a^2 into (b-a)(b+a) then multiplying row 2 times (b+a) and subtracting from row three but then the last element in row three would be (c^2-a^2) - (c-a)(b+a)... Taking the determinant here yields nothing productive. Stuck again.. Thanks very much for the help so far though.[/code]

What do you get as determinant? - write it out.

 

[jman2807]

Ah ha... It was a pain but once i got it factored it did turn out to be (c-a)(b-a)(c-b)

Thank you....