Write down a linear operator $f: \mathbb{R}^4\to\mathbb{R}^4$ whose minimal polynomial is $m_f(t)=t^3-t^2$
I know that since in $\mathbb{R}^4$ we will have a characteristic polynomial $p_f(t)=t^3(t-1)$ or $p_f(t)=t^2(t-1)^2$.
But we have to check $m_f(f)=0$ for our selected matrix which would involve a lot of matrix multiplication, I was wondering if there was an easier way to do this without having to plug our matrix into the minimal polynomial and writing pages of matrix multiplication for each test.
Or what would be an example of a matrix that I could use, and how did you decide on that matrix.
Any help and hints would be greatly appreciated.
For example I had:
$$A=\begin{pmatrix}0&0&0&0\\0&0&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}$$
I compute the characteristic polynomial:
$p_f(t)=det(A-tI)=det\begin{pmatrix}-t&0&0&0\\0&-t&0&0\\0&0&1-t&0\\0&0&0&1-t\end{pmatrix}=t^2(1-t)^2$, now how would I go about checking if it was minimal.
It would take ages to check and this was only a 3 mark question, there must be a quicker way, or a more obvious answer.