Eigenvalues of a $2\times2$ matrix 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.  

*$A^3=\frac{a^3-b^3}{a-b}A-ab(a+b)I$   

*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?   

Hint: For option $4$, construct a matrix $A$ with the diagonal elements $1$ and $-1$. Then check $A^2$ = identity matrix.
