Matrix proof

[Gninnahc]

Hello all...

Ok so Let A =
[ 1 2]
[-1 -2]

I want to show that A^2n = -A. With matrix computation I tried 2, 3 & 4 as test cases.
It works because 2 * any positive number gives you an even number...

So A^2(2) = A^4 =
[-1 -2]
[ 1 2 ]

-A =
[-1 -2]
[ 1 2 ]

therefore A^2n = -A are equal when n = 2. How do I show this for ALL positive integers in a proof?
Thanks and any hints would be appreciated.

 

[daon2]

Induction

\(\displaystyle A^{2(n+1)} = A^{2n}\cdot A^2 = (-A)(-A) = A^2 = -A\).

 

[Gninnahc]

I'm currently trying to prove by induction.

 

[stapel]

I'm currently trying to prove by induction.

And you've been shown the "n + 1" step. Where are you stuck?

 

[Gninnahc]

And you've been shown the "n + 1" step. Where are you stuck?

Yea, I didn't refresh the page before I posted... :lol:... not stuck anymore.

Thanks daon2.