Question

State if the following statements are true and which are false? Justify your answer with a
short proof or a counterexample.

If the characteristic polynomial of a linear transformation is (x-1)(x-2), its
minimal polynomial is x-1 or x-2.

Answer 1

Answer on Question #44922 – Math - Linear Algebra

Problem.

State if the following statements are true and which are false? Justify your answer with a short proof or a counterexample.

If the characteristic polynomial of a linear transformation is $(x-1)(x-2)$, its minimal polynomial is $x-1$ or $x-2$.

Solution.

The statement is false.

Suppose that $A = \begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix}$ is a matrix of linear transformation. The characteristic polynomial of a linear transformation is $(x - 1)(x - 2)$, as

$$\det \begin{bmatrix} 1 - x & 0 \\ 0 & 2 - x \end{bmatrix} = (1 - x)(2 - x) = (x - 1)(x - 2).$$

$x - 1$ isn't minimal polynomial of $\begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix}$, as $\begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix} - \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} \neq \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$.

$x - 2$ isn't minimal polynomial of $\begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix}$, as $\begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix} - 2 \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} -1 & 0 \\ 0 & 0 \end{bmatrix} \neq \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$.

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