<s>[INST] Find a $3\times 3$ matrix $A\not = I_3$ such that $A^3 = I_{3}$ Use the correspondence between matrices and linear transformation to find find a $3\times 3$ matrix $A$ such that $A^3 = I_{3}$ and find an $A$ matrix that is not $I_{3}$
Where $I_{3}$ is the identity matrix:
$$I_{3}=
  \left[ {\begin{array}{ccc}
   1 & 0 & 0\\
   0 & 1 & 0\\
   0 & 0 & 1\\
  \end{array} } \right]$$
I was tried with the following $A$ matrix:
$$A=
  \left[ {\begin{array}{ccc}
   1 & 0 & 0\\
   0 & 1 & 0\\
   0 & 0 & 1\\
  \end{array} } \right]$$
and when I multiply $A \times A \times A$ I got the same matrix as $I_{3}$.
And to find a matrix $A$ that is not equal to $I_{3}$ I can take any $A$ matrix that is not: 
$$A=
  \left[ {\begin{array}{ccc}
   1 & 0 & 0\\
   0 & 1 & 0\\
   0 & 0 & 1\\
  \end{array} } \right]$$
but I think that the exercise is expecting something else using linear transformations.
Sorry I realised that $A$ cannot be equal to $I_{3}$
 [/INST] Try
\begin{align}
A =
\begin{bmatrix}
0 & 1 & 0\\
0 & 0 & 1\\
1 & 0 & 0
\end{bmatrix}
\end{align}
</s>