Drawing conclusions from a characteristic polynomial

Question

I was given a characteristic polynomial and was asked to draw some conclusions from it, but i need some help with figuring something out. My polynomial is this : $\lambda^4 -3\lambda^3 +\lambda^2 +3\lambda -2$. I was able to simplify it to this : $(\lambda-1)^2(\lambda+1)(\lambda-2)$. So i have 4 eigenvalues $\lambda=1$ (with a 2 degree multiplicity), $\lambda=-1$, $\lambda=2$. The conclusion i have drawn so far is that it can be inverted since the determinant is not $0$. How would i find the rank of the original matrix and how would i answer to whether this matrix could be diagonalizable?

Answer 1

The rank of a matrix is determined by the number of its non-zero singular values (or in the case of square matrices, the number of non-zero eigenvalues). Since the matrix has exactly $4$ non zero eigenvalues, its rank becomes $4$, but nonetheless nothing can be said about its diagonalizability as the following two matrices yield to the same characteristic polynomial$$A=\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&-1&0\\0&0&0&2\end{bmatrix}$$ $$B=\begin{bmatrix}1&1&0&0\\0&1&0&0\\0&0&-1&0\\0&0&0&2\end{bmatrix}$$while $A$ is diagonalizable and $B$ is not.