Find the minimal polynomial for the given matrix. The first part is trivial. How to go about the second part?


You can check that
$$A^2=\begin{pmatrix}
0 &c &ac\\
0&b&ab+c\\
1&a&a^2+b
\end{pmatrix}
$$
Suppose that the minimal polynomial is $m(x) = x^2+\alpha x+\beta$. Then $m(A) = 0$. Notice that 
$$(m(A))_{11} = 0+0+\beta=0\Rightarrow \beta=0$$
$$(m(A))_{21} = 0+\alpha+0=0\Rightarrow \alpha=0$$
Thus, $A^2=0$ which is clearly false.
By a simpler argument you can conclude that $m(x)$ is not a linear polynomial either. Hence, $m(x)$ must be at least of degree 3. But the characteristic polynomial is of degree 3 and we are done.
