I am looking for some help on a little matrix/determinant problem please?

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The discussion focuses on calculating the determinant of a 3x3 matrix, specifically the matrix defined as: | 1 1 -1 |, | a -b c |, | a² b² -c² |. The correct determinant is established as (a+b)(a+c)(c-b). The user struggles with the calculation but eventually realizes that simplifying the matrix using column operations leads to the correct answer. The final determinant calculation involves recognizing the structure of the matrix and applying properties of determinants effectively.

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jeebs
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i get the feeling that its going to be hard to type this out but here goes. sorry about the dots, it was the only way.
i got to show that the determinant

|..1...1...-1...|
|..a...-b...c...| = (a+b)(a+c)(c-b).
|a^2...b^2...-c^2.|

however, i cannot quite get the answer to this. there is definitely one of these questions on my exam on friday and i just cannot ever reach the finish line with these ones.

here is as far as i get with this particular one:

|..1...1...-1..|
|..a...-b...c..|
|a^2...b^2...-c^2|

=

|..1-1...1...1-1...|
|..a+b...-b...a+c...| (took column 2 from column 1 - determinant unchanged)
|a^2+b^2...b^2...a^2-c^2..|

=

|...0....1....0...|
|...a+b...-b...a+c...| (neatened things up a bit)
|(a+b)(a-b)...b^2...(a+c)(a-c)|

=

|..0...1...0...|
1/(a+b)(a+c)|..1...-b...1...| (multiplied column 1 by 1/(a+b) & column 2 by 1/(a+c)
|a-b...b^2...a-c.| - therefore multiplied determinant by same amount)

= (b-c)/(a+b)(a+c) when that simplified determinant is calculated.

ive tried messing around a few ways but this is the closest i have got to the answer, as in this attempt i at least got the (a+b) and (a+c) parts.

can somebody show me the right way to do this please, and most importantly explain it?

thanks.
 
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|...0....1....0...|
|...a+b...-b...a+c...| (neatened things up a bit)
|(a+b)(a-b)...b^2...(a+c)(a-c)|
Direct calculation of the determinant from the above gives the correct answer - since the first row is 0 1 0, the determinant is given by 2x2 determinant from rows 2 and 3 and columns 1 and 3, with a sign reversal.
 
oh, thank you so much, i feel like an idiot, i have been on AGES trying to figure these out, i feel like an idiot lol
 

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