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?

1. $A^2$ has distinct eigen values.
2. $A^3=\frac{a^3-b^3}{a-b}A-ab(a+b)I$
3. Trace of $A^n$ is $a^n+b^n$ for every positive integer n.
4. $A^n$ 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?

• You haven't provided any proof for option (2), despite your claim. You can finish it off easily by using Cayley-Hamilton theorem. – user1551 May 3 '17 at 6:23
• It is just calculation by Cayley-Hamilton Theorem. – user159480 May 3 '17 at 6:39

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