linear algebra - How to conclude that the minimal polynomial is the characteristic?

I am given the following matrix

$$A=\begin{bmatrix} 0 & 0 & 4 & 1\\ 0& 0 & 1 & 4\\ 4 & 1 & 0 &0\\ 1 & 4 & 0 & 0 \end{bmatrix}$$
And I have to find the minimal polynomial of the matrix. The characteristic polynomial is $$K(\lambda)=-(\lambda -5)(\lambda +5)(\lambda-3)(\lambda +3)$$
The minimal polynomial $m(\lambda)$ divides the characteristic polynomial. I know that the characteristic polynomial is the minimal, but how do i eliminate the possibilities of the linear, quadratic and qubic factors in the polynomial. When do i know the minimal is actually the characteristic polynomial?