Exercise 7.3.2

Let $A$ be the given matrix. Find the eigenvalues ${\lambda }_i$ of $A$. Then the minimal polynomial should be of the form ${\prod <!--\; FUNCTION\; APPLICATION\; -->}_i{(t-{\lambda }_i)}_i^r$. Try all possible $r_i$’s. Another way is to compute the Jordan canonical form.

1.

The eigenvalues are $1$ and $3$. So the minimal polynomial must be $(t-1)(t-3)$.

2.

The eigenvalues are $1$ and $1$. So the minimal polynomial could be $(t-1)$ or ${(t-1)}^2$. Since $A-I\ne O$, the minimal polynomial must be ${(t-1)}^2$.

3.

The Jordan canonical form is

$$\left(\begin{array}{ccc}\hfill 2\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right).$$

So the minimal polynomial is $(t-2){(t-1)}^2$.

4.

The Jordan canonical form is

$$\left(\begin{array}{ccc}\hfill 2\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 2\hfill \end{array}\right).$$

So the minimal polynomial is ${(t-2)}^2$.