let $\lambda, \mu$ be distinct eigenvalues of a $2 \times 2 $ matrix $A$.Then which of the following statements must be true?

Question

I was thinking about the following problem..

let $\lambda, \mu$ be distinct eigenvalues of a $2 \times 2 $ matrix $A$.Then which of the following statements must be true?

a. $A^2$ has distinct eigenvalues.

False .Counter Example: $\begin{pmatrix} 1 &0 \\ 0 & -1 \end{pmatrix}$

b. $A^n$ is not a scalar multiple of identity for any integer $n$

False .Counter Example: $\begin{pmatrix} 1 &0 \\ 0 & -1 \end{pmatrix}$ as this gives $A^2=1.I$ where $I=\begin{pmatrix} 1 &0 \\ 0 & 1 \end{pmatrix}$ and so second statement is violated for $n=2$.

Am I right? Please feel free to comment. Thanks and regards to all.

Answer 1

From the comment mentioned above by @5xum and to save this question from being unanswered,I just pick up my two counter-examples for the answer of this particular question.

a. $A^2$ has distinct eigenvalues.

The above statement is False .Counter Example: $\begin{pmatrix} 1 &0 \\ 0 & -1 \end{pmatrix}$

b. $A^n$ is not a scalar multiple of identity for any integer $n$

The above statement is False .Counter Example: $\begin{pmatrix} 1 &0 \\ 0 & -1 \end{pmatrix}$ as this gives $A^2=1.I$ where $I=\begin{pmatrix} 1 &0 \\ 0 & 1 \end{pmatrix}$ and so second statement is violated for $n=2$.