Prove that the matrix
$$A = \begin{bmatrix} 0 & 1 & \\ \ddots & & \ddots \\ & \ddots & & \ddots \\ & & \ddots & & \ddots\\ & & & 0 & & 1\\ -a_{0} & -a_{1} & \dots & \dots & \dots & -a_{n-1}\end{bmatrix}$$
is non-derogatory. I am not sure how would I do this. Non-derogatory matrix means that the minimal polynomial equals the characteristic polynomial but I am not sure how to this one.