Question

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

Answer 1

The correct answer is 25
$\therefore A^2 = \begin{bmatrix} 1+i & 1 \\\ -i & 0 \end{bmatrix} \begin{bmatrix} 1+i & 1 \\\ -1 & 0 \end{bmatrix} = \begin{bmatrix} i & 1+i \\\ 1-i & -i \end{bmatrix}$
$A^4 = \begin{bmatrix} i & 1+i \\\ 1-i & -i \end{bmatrix} \begin{bmatrix} i & 1+i \\\ 1-i & -i \end{bmatrix} = l$
So A5 = A, A9 = A and so on.
Clearly n = 1, 5, 9, ….., 97
Thereore , number of values of n = 25