True or false and explain why?: a matrix with characteristic polynomial $\lambda^3 -3\lambda^2+2\lambda$ must be diagonalizable.
First I found the lambda's that make this zero (eigenvalues) and got $0, 1, 2$ but I don't know if having $0$ as an eigenvalue means that the matrix is not diagonalizable? I know that a matrix has $0$ as an eigenvalue if it is not invertible, but I don't know if a matrix needs to be invertible to be diagonalizable? Also if a matrix has complex eigenvalues does that also mean it cannot be diagonalizable?