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?
Best Answer
The characteristic polynomial splits as follows $P_M(\lambda)=\lambda(\lambda-1)(\lambda-2)$. The matrix is $3\times 3$ (the degree of the characteristic polynomial) and has three distinct eigenvalues therefore it is diagonalisable.