Let $a,b$ be distinct eigen values of a $2\times2$ matrix $A.$Then which of the following statement is true?
- $A^2$ has distinct eigen values.
- Trace of $A^n$ is $a^n+b^n$ for every positive integer n.
- $A^n$ is not a scalar multiple of identity matrix for any positive integer n.
I think first option is wrong because if $1,-1$ are distinct eigenvalues of $A$ but $A^2$ has eigenvalues $1,1$. Third option is correct as we have result. I tried second option. Second is also right. For fourth option, characteristic equation implies $A^n$ cannot be a scalar multiple of identity matrix for any positive integer $n$. Is it correct?