Finite field/Invertible matrix/Z mod 3/0120/Order/Exercise/Solution

We have

${}{\begin{pmatrix}0&1\\2&0\end{pmatrix}}^{2}={\begin{pmatrix}0&1\\2&0\end{pmatrix}}{\begin{pmatrix}0&1\\2&0\end{pmatrix}}={\begin{pmatrix}2&0\\0&2\end{pmatrix}}\,,$

${}{\begin{pmatrix}0&1\\2&0\end{pmatrix}}^{3}={\begin{pmatrix}2&0\\0&2\end{pmatrix}}{\begin{pmatrix}0&1\\2&0\end{pmatrix}}={\begin{pmatrix}0&2\\1&0\end{pmatrix}}\,,$

and

${}{\begin{pmatrix}0&1\\2&0\end{pmatrix}}^{4}={\begin{pmatrix}2&0\\0&2\end{pmatrix}}{\begin{pmatrix}2&0\\0&2\end{pmatrix}}={\begin{pmatrix}1&0\\0&1\end{pmatrix}}\,,$

therefore, the order is ${}4$.

To the exercise