I found a problem in a linear algebra book which is as follows:
Give an example of a linear operator on $\mathbb C^4$ whose characteristic and minimal polynomial both equal to $X(X-1)^2(X-3)$.
I think I can solve this by using the knowledge of Jordan canonical form.Although it is easy to construct a linear operator with the given characteristic polynomial but it is not easy to guarantee that this is also the minimal polynomial.So,I can construct the operator $T$ whose Jordan canonical form is,
$\begin{pmatrix}0 & & & \\ &1 & &\\ &1 &1 & \\ & & & 3\\\end{pmatrix}$
Then the minimal polynomial is $X^{r_1}(X-1)^{r_2}(X-3)^{r_3}$ where $r_i$ is the size of the largest Jordan block corresponding to eigenvalue $\lambda_i$.I think this is the easiest way to solve such problems.Another approach is motivated from the comments that is to use companion matrix.For companion matrix characteristic and minimal polynomials are same.Notice that the polynomial $X(X-1)^2(X-3)=X^4-5X^3+7X^2-3X$ and the companion matrix of this polynomial is $\begin{pmatrix} & & & 0\\1& & & 3\\ &1 & &-7\\ & & 1& 5\\\end{pmatrix}$,so this operator will also work.Is my approach correct?