For non-constant polynomial $f(t)$, there is a linear transformation $T$ such that $\phi_T(t) = (-1)^n f(t)$

Question

I want to prove the following theorem

Let $f(t)$ be a non-constant monic polynomial of degree $n$. Then there exists a linear operator $T$ on $n-$dimensional space $V$ with the characteristic polynomial $\phi_T(t)= (-1)^n f(t)$.

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How one can prove this? I know characteristic polynomial of a linear operator $T$ is given $\phi_T(t)=\det(T-t I)$ and since $T$ is $n\times n$ matrix it has the form of $(-1)^n (t^n+ \cdots)$, so Naively for any linear operator $T$ I can find the non-constant monic polynomial of degree $n$.

Is my approach valid?