Imagenary cube root Problems

• May 3rd 2009, 10:11 PM
zorro
Imagenary cube root Problems
If $\displaystyle w$ is an imagenary cube root of unity , then find the value of

$\displaystyle \begin{vmatrix} 1+w & w^2 & -w \\ 1+w^2 & w & -w^2 \\ w^2+w & w & -w^2 \end{vmatrix}$
• May 4th 2009, 01:42 AM
Gamma
did you type it right? that bottom 2x2 minor has rows the same, is that correct?

If so, just do it, i recommend cofactor down the first column to take advantage of that being the same.

the first term is 0.

second is
$\displaystyle -(1+w^2)[-w+w^2]$
third is
$\displaystyle (w^2+w)[-w+w^2]$

$\displaystyle (-1-w^2 +w^2 +w)[-w+w^2]=(-1+w)[-w+w^2]=1-w^2-w^2+w=-2w^2+w+1$
But these are cube roots of unity, $\displaystyle w^2+w=-1 \Rightarrow w=-1-w^2$

$\displaystyle -2w^2+w+1= -2w^2 (-1 - w^2) + 1= -3w^3=-3$
• Dec 14th 2009, 01:16 PM
zorro
I didnt get the last part
Quote:

Originally Posted by Gamma
did you type it right? that bottom 2x2 minor has rows the same, is that correct?

If so, just do it, i recommend cofactor down the first column to take advantage of that being the same.

the first term is 0.

second is
$\displaystyle -(1+w^2)[-w+w^2]$
third is
$\displaystyle (w^2+w)[-w+w^2]$

$\displaystyle (-1-w^2 +w^2 +w)[-w+w^2]=(-1+w)[-w+w^2]=1-w^2-w^2+w=-2w^2+w+1$
But these are cube roots of unity, $\displaystyle w^2+w=-1 \Rightarrow w=-1-w^2$

$\displaystyle -2w^2+w+1= -2w^2 (-1 - w^2) + 1= -3w^3=-3$

I didnt get the last part
$\displaystyle -2w^2+w+1= -2w^2 (-1 - w^2) + 1= -3w^3=-3$

could u please explain how u got this?????
• Dec 19th 2009, 07:52 PM
zorro
i have some doubt
Quote:

Originally Posted by zorro
I didnt get the last part
$\displaystyle -2w^2+w+1= -2w^2 (-1 - w^2) + 1= -3w^3=-3$

could u please explain how u got this?????

-Firstly the $\displaystyle -3w^3$ shouldnt it be $\displaystyle -3w^2$

-Secondly where did the $\displaystyle w^2$ go and how did $\displaystyle -3w^2 \ = \ -3$
• Dec 19th 2009, 08:29 PM
zorro
-Firstly the $\displaystyle -3w^3$ shouldnt it be $\displaystyle -3w^2$
-Secondly where did the $\displaystyle w^2$ go and how did $\displaystyle -3w^2 \ = \ -3$