1. ## Matrices........

Hello!

I have this to "decipher":

If 1 + ALPHA + ALPHA^2 = 0, show that ALPHA^3 = 1.

Many thanks!

2. ## Re: Matrices........

I'm not sure why your topic says "matrices", but if $\displaystyle \displaystyle \alpha^3 = 1$ then

\displaystyle \displaystyle \begin{align*} \alpha^3 - 1 &= 0 \\ \alpha^3 - 1^3 &= 0 \\ (\alpha - 1)(\alpha^2 + \alpha + 1) &= 0 \\ \alpha^2 + \alpha + 1 &= 0 \textrm{ or }\alpha - 1 &= 0 \end{align*}

3. ## Re: Matrices........

Originally Posted by Biz
Hello!

I have this to "decipher":

If 1 + ALPHA + ALPHA^2 = 0, show that ALPHA^3 = 1.

Many thanks!
If '1' means 'unity matrix' and '0' means 'null matrix' then is...

$\displaystyle 1 + A + A*A=0 \implies A*A = -1 - A \implies A* A* A = - A - A* A = -A +1 + A = 1$

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

4. ## Re: Matrices........

Thank you!

Just curious but the font used looks very similar to a set of workbooks that we have been advised to study called HELM.

Might just be a coincidence!