Find a $3\times 3$ matrix $A\not = I_3$ such that $A^3 = I_{3}$

Question

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 $I_{3}$

Answer 1

Try \begin{align} A = \begin{bmatrix} 0 & 1 & 0\\ 0 & 0 & 1\\ 1 & 0 & 0 \end{bmatrix} \end{align}

Answer 2

Let $e_1,e_2,e_3$ be the standard basis vectors. We can look at the linear transformation, T, that maps $e_1\mapsto e_2$, $e_2\mapsto e_3$ and $e_3\mapsto e_1$. Then it is clear that if we apply T to the standard basis vectors 3 times, we get the identity transformation.

Now you have to convert the transformation T into matrix form - can you do this?

Extra: if you are concerned with finding $n\times n$ matrices such that $A^k = I_n$ we can find permutations of $\{e_1,\dots,e_n\}$ of order $k$ (or order that divides $k$). Then we can construct the matrix using the correspondence between linear maps and matrices (and the fact that linear transformations are completely determined by the basis vectors). You will find that for large $n$, checking this is much easier than multiplying out the matrices directly.