Does a zero eigenvalue mean that the matrix is not invertible regardless of its diagonalizability? If the matrix $A$ is diagonalizable, then we know that its similar diagonal matrix $D$ has determinant $0$, so the matrix $A$ itself is invertible? However, if $A$ not diagonalizable, how are we sure that the matrix $A$ which has $0$ as eigenvalue is not invertible?
Here I have another confusion, does the degree of the characteristic polynomial  determine the size of matrix. i.e. $\lambda (\lambda+2)^3 (\lambda-1)^2$ has $6\times 6$ matrix?    

Suppose $M$ is an invertible matrix, with nonzero eigenvector $v$ corresponding to the eigenvalue $0$.  Then we would have
$$v = M^{-1}Mv = M^{-1}(0v) = 0$$
But $v \ne 0$.  This shows that if $0$ is an eigenvalue of $M$, $M$ cannot be invertible.
