Let S=nN:pmatrix0&i1&0pmatrix^npmatrixa&b&dpmatrix=pmatrixa&b&dpmatrix , a,b,c,dR, where i…

Question

Let $S=\left\{n\in\mathbb{N}:\begin{pmatrix}0&i\1&0\end{pmatrix}^n\begin{pmatrix}a&b\c&d\end{pmatrix}=\begin{pmatrix}a&b\c&d\end{pmatrix}\ \forall\, a,b,c,d\in\mathbb{R}ight\}$, where $i=\sqrt{-1}$. Then the number of $2$-digit numbers in the set $S$ is

Answer: 11

Accepted Answer

Let $A=\begin{pmatrix}0&i\1&0\end{pmatrix}$; the condition holds iff $A^n=I$.

$A^2=\begin{pmatrix}i&0\0&i\end{pmatrix}=iI$, so $A^8=(A^2)^4=i^4I=I$. In general, $A^n=I$ iff $8\mid n$.

Two-digit multiples of $8$: $16,24,\dots,96$, giving $\dfrac{96-16}8+1=11$ values.

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