$A$ be a complex $3\times 3$ matrix such that $A^3=-I$

Question

Let $A$ be a complex $3\times 3$ matrix such that $A^3=-I$, then we need to find out which of the following statements are correct?

1. $A$ has three distinct eigenvalues;

2. $A$ is diagonalizable over $\mathbb{C}$;

3. $A$ is triangulizable over $\mathbb{C}$;

4. $A$ is non singular.

Wll, from minimal polynomial approach I have deduced that all are correct as minpoly has to be $x^3+1$ am I right?

Answer 1

Take $A=-I$. Then $A^3 = -I$, but $A$ does not have distinct eigenvalues.

Hint:

The minimal polynomial of a matrix $A$ may be defined as the polynomial of smallest degree that is satisfied by A and has highest coefficient equal to 1.

Is $x^3 +1$ really the polynomial of smallest degree?

Answer 2

Since 1 was answered for you, here is a hint for 4: $\det (A^3) = (\det A)^3$.

Once you know that, can Jordan blocks of order greater than $1$ vanish or become diagonal when you compute their powers? This should answer 2 and 3.