A good property of trace 0 matrices

I found out an interesting property of a trace $0$ matrix.Suppose $A$ be a $n\times n$ matrix whose trace is $0$. Then $A^2$ will be a scalar matrix (that is $A^2$ is of the form $\lambda I_n$, where $\lambda$ is any real constant and $I_n$ is the identity matrix ) iff $n=2$.But however, this result does not hold true for $n \ge 3$.For $n=2$ it is simple to prove if we assume the form of the matrix and just use the restriction $\operatorname{tr}(A)=0$.But how to approach this using eigenvalues? Also, how can we infer that this result does not hold for higher $n$ using eigenvalues?