Let $A$ be a complex $3 \times 3$ matrix with $A^3 = -I$.

Question

Let $A$ be a complex $3 \times 3$ matrix with $A^3 = -I$. Which of the following statements are correct?

1. $A$ has three distinct eigenvalues.
2. $A$ is diagonalizable over $\mathbb{C}$.
3. $A$ ls triangularizable over $\mathbb{C}$.
4. $A$ is non-singular.
   

If we take $A = -I$, then $A^3 = -I$ and has eigenvalues as $-1, -1, -1$. so option $1$ is incorrect. Since $A$ is not equal to $0$, $A$ is non-singular and also option $2$ and $3$ are correct. Hence option $2, 3$, and should be options. Is it correct?

Answer 1

Yes you answer 1) is correct.

2) Using the Julien's hint, let $p(x)=x^3+1$ then this polynomial with distinct roots in $\mathbb C$ annihilates the matrix $A$ and then this matrix is diagonalizable over $\mathbb C$.

3) Every matrix is trigonalizable over $\mathbb C$

4) We have $\det(A)^3=\det(A^3)=\det(-I)=-1\neq 0$ so $A$ is invertible.

Answer 2

Diagonalizable means the matrix has three distinct eigenvectors.

We cannot infer that the minimal polynomial is $\lambda^3+1=0$.

So, I think 2) might be incorrect.

For 1) 3) 4) I agree with Sami Ben Romdhane.