Is There a Mistake in My Proof for the Identity of Pauli Spin Matrices?

[InGaAsP]

Homework Statement

prove the idendity $$\sigma \times \sigma =i \sigma$$ where $$\sigma$$ is Pauli Spin matrices

Homework Equations

The Attempt at a Solution

This is how I did..and I am getting
2$$i \sigma$$ instead of $$i\sigma$$.

http://i146.photobucket.com/albums/r273/soorajr/paulimatrix.jpg

This question was asked twice for the university exam, and during both times, they asked us to prove $$\sigma \times \sigma =i \sigma$$.
Did i make any mistake? or the examiner was wrong?
Thanks in advance.

 

[Galileo]

You wrote (for instance) $\sigma_y\sigma_z-\sigma_z\sigma_y=i\sigma_x$
but then later when you get the same expression for the x-component you enter
$i\sigma_x+i\sigma_x$, which is where your factor 2 comes from.
Check the middle line in your calculation of the determinant again

 

[InGaAsP]

You wrote (for instance) $\sigma_y\sigma_z-\sigma_z\sigma_y=i\sigma_x$
but then later when you get the same expression for the x-component you enter
$i\sigma_x+i\sigma_x$, which is where your factor 2 comes from.
Check the middle line in your calculation of the determinant again

Thanks Galileo for ur reply

the results i used were$$\sigma_x\sigma_y=-\sigma_y\sigma_x=i\sigma_z$$
i changed x,y,z cyclicly, and hence reached at 2$$i \sigma$$
do u think I am supposed to get only i$$\sigma$$ instead of 2i$$\sigma$$
thanks in advance

 

[nrqed]

Homework Statement

prove the idendity $$\sigma \times \sigma =i \sigma$$ where $$\sigma$$ is Pauli Spin matrices

Homework Equations

The Attempt at a Solution

This is how I did..and I am getting
2$$i \sigma$$ instead of $$i\sigma$$.

http://i146.photobucket.com/albums/r273/soorajr/paulimatrix.jpg

This question was asked twice for the university exam, and during both times, they asked us to prove $$\sigma \times \sigma =i \sigma$$.
Did i make any mistake? or the examiner was wrong?
Thanks in advance.

Yes, there is a mistake in the question. There really should be a factor fo 2 there.

 

[Galileo]

Indeed, your first lines should be $\sigma_y\sigma_z-\sigma_z\sigma_y=2i\sigma_x$ etc.