Question

How should I find this determinant? Image attached.

Answer 1

Whoever set this on you is a sadist... o.O

Answer 2

Have you tried evaluating it normally? (that must be tough...)

Answer 3

I tried to bring it to the Vandermonde form, whose result is simple.

Answer 4

Suffice it to say, matrices are not my specialty... let's bring in some heavy-hitters...
@hartnn

Answer 5

also tried to substract : Line2 - a*Line1, Line3 - a*Line2, but I can't arrive at the result of the wolfram alpha which is> \[abc \times (a-b)^{2} \times (a-c)^{2} \times (b-c)^{2}\]

Answer 6

I will revise my work because I might have made some mistakes in terms of simple calculus... overwhelmed by its huge amount. :)

Answer 7

copy the first 2 columns at the end, then multiply the diagonals
\[\begin{vmatrix}
a&b&c&|&a&b\\d&e&f&|&d&e\\g&h&i&|&g&h\end{vmatrix}\]

Answer 8

dont bother working out all the multiplications since they might simplify out in the end

Answer 9

mm, you mean using the Rule of Sarrus.

Answer 10

i got no idea what the name is :)

(123+234+243)-(333+441+522)

i just used the exponents there to keep it cleaner

Answer 11

How do you augment a matrix, @amistre64 ?

Answer 12

augment is just running elementary row operations until you get to a reduced form

Answer 13

No, I mean, how did you typeset an augmented matrix in TeX :)

Answer 14

http://en.wikipedia.org/wiki/Rule_of_Sarrus . take a look at the main picture.

Answer 15

i only know how to code up the matrix forms using \begin{} \end{} syntaxes

Answer 16

yeah, that rule :)

Answer 17

Oh well, I doubt I'll use matrices that much anyway :)

Answer 18

ok, thanks for the tip. :)

Answer 19

123 + 234 + 234 - 333 - 441 - 522

i dont see a nice way to simplify that except for expanding them out in a painstaking manner now

Answer 20

yeah, the exercises in this book are quite rude and I got used to the idea.

Answer 21

it might help to know that:\[(a+b+c)(x+y+z)=ax+ay+az+bx+by+bz+cx+cy+cz\]
when thats a perfect square you get
\[(a+b+c)^2=aa+ab+ac+ba+bb+bc+ca+cb+cc\\~~~~~~~~~~~~~~~~~=a^2+b^2+c^2+2ab+2ac+2bc\]