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Number of n for which a power condition such as A^n = A holds (powers repeat in a cycle)

This type of question, asked 3 times: 2021 Paper 14, 28 Jun 2022, 25 Jul 2022. All 3 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    2021 Paper 14

    56
    Let S={n∈N,[0ii0]n[abcd]=[abcd]∀a,b,c,d∈R]S=\left\{ n\in N,\begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix}^{n}\begin{bmatrix} a & b \\ c & d \end{bmatrix}=\begin{bmatrix} a & b \\ c & d \end{bmatrix}\forall a,b,c,d\in R \right\rbrack. Find number of 2-digit numbers in 5

  2. Question 2

    28 Jun 2022

    177
    Let A=(1+i1−i0)A =\begin{pmatrix} 1 + i & 1 \\ - i & 0 \end{pmatrix} where i=−1i =\sqrt{- 1}. Then, the number of elements in the set {n∈{1,2,…,100}:An=A}\left\{ \mathbf{n \in \{ 1,2,\ldots,100\}:}A^{n}= A \right\} is

  3. Question 3

    25 Jul 2022

    81
    Let A=[2−1−110−11−10]A =\begin{bmatrix} 2 & - 1 & - 1 \\ 1 & 0 & - 1 \\ 1 & - 1 & 0 \end{bmatrix} and B=A−IB = A - I. If ω=3i−12\omega =\frac{\sqrt{3}i - 1}{2}, then the number of elements in the set {n∈{1,2,…,100}:An+(ωB)n=A+B}\left\{ \mathbf{n \in \{ 1,2,\ldots,100\}:}A^{n}+ (\omega B)^{n}= A + B \right\} is equal to