With DonAntonio's help Composition of linear maps. I managed to find $t^4 = t^2 +4(t^2- id)$ , $t^6 = t^2 + 4(4+1) (t^2 -id)$ and $t^8 = t^2 + 4 ( 1+ 4 +4^2)(t^2 -id) $
So now I want to prove it true for some $n $
$t^{2n} = t^2 + 4(1+4+4^2+...+4^{n-2}) (t^2 -id)$
My immediate thought was Proof by induction and started as such:
Assume it true for some $n=k$ then
$t^{2k+2} = t^{2k} + t^2$
by the induction hypothesis
$t^{2k +2} = t^2[t^2 +4 (1+4+4^2+...+4^{k-2}) (t^2 -id)] $
However whatever I do , it always leads to a messy expansion that would probably take hours to solve and simplify
Can anyone suggest a quicker way?
First, I think you made a sign error: $t^4 = t^2 + 4(t^2-I)$, not $t^2+I$. Second, I think your formula will be easier to understand, and the induction will be simpler to prove if you expand the powers you computed explicitly and extrapolate to an explicit formula for $t^{2n}$.