Show that there are infinitely many matrices $X \in M_2(C)$ that simultaneously check properties:
$X^2 \neq I_2$
$X^2 \neq -I_2$
$X^4=I_2$
That's what I wrote: $\det X=\pm 1$ and $tr X=t \neq 0$
Case 1: $\det X=1$
With Cayley-Hamilton $X^2=tX-I_2$ and by multiplying with X 2 times it results $X^4=t(t^2-2)X-(t^2-1)I_2=I_2$. Using the trace we obtain $t^2(t^2-2)-2(t^2-1)=2$ and by calculus $t^2=4$ or $t=\pm 2$. But if I put $t$ in the recurrence we have $X=\pm I_2$ and $X^2=I_2$, which is not true.
Case 2: $\det X=-1$
With Cayley and the same idea $X^4=t(t^2+2)X+(t^2+1)I_2=I_2$. Using the trace, $t^2+4=0$ or $t=\pm 2i$. But next we have $X=iI_2$ and $X^2=-I_2$, false.
Please help me!
How about matrices with eigenvalues $1$ and $i$? Conjugates of $$\pmatrix{1&0\\0&i}?$$ Are there infinitely many conjugates of this matrix?