$3 \times 3$ Diagonal matrix as the square of an $3 \times 3$ non diagonal matrix

Question

It is easy to see if the matrix is $2 \times 2$, by just considering a system of equation with $4$ equations.

However, for a $3 \times 3$ matrices, a system of equation with $9$ equations might not be a good approuch to solve this problem.

Is there any other way to attack this problem? Or is this really solvable or does is really exists ?

If you have an example. Pls send one. Thank you very much

Answer 1

Hint Since you've already found an example $A$ of a nondiagonal $2 \times 2$ matrix satisfying $A^2 = I_2$, consider the (hence nondiagonal) $3 \times 3$ matrix $$\pmatrix{A&\\&1}.$$