Let A=pmatrix1+i&1-i&0pmatrix where i=-1.

Question

Let $A=\begin{pmatrix}1+i&1\-i&0\end{pmatrix}$ where $i=\sqrt{-1}$. Then the number of elements in the set $\{n\in\{1,2,\dots,100\}: A^n=A\}$ is\_\_\_\_

Answer: 25

Accepted Answer

$A^2=\begin{pmatrix}i&1+i\1-i&-i\end{pmatrix}$, and $A^4=I$ (direct computation).

So $A^n=A$ whenever $n\equiv1\pmod4$: $n=1,5,9,\dots,97$.

Number of such $n$ in $\{1,\dots,100\}$: $\dfrac{97-1}4+1=25$.

Solutions · 1