A square matrix $\displaystyle A$ (of some size $\displaystyle n \times n$) satisfies the condition $\displaystyle A^2 - 8A +15I = 0.$

(a) Show that this matrix is similar to a diagonal matrix.

(b) Show that for every positive integer $\displaystyle k \geq 8 $ there exists a matrix $\displaystyle A$ satisfying the above condition with $\displaystyle tr(A) = k$.

I have little or no idea where to go with this problem. I know that for (a), $\displaystyle A^2 - 8A$ is obviously a diagonal matrix $\displaystyle (-15I)$, but am not sure if this is of any use. I also am not sure if I can simply say $\displaystyle A^2 - 8A + 15I = 0$, so $\displaystyle (A - 5I) (A - 3I) = 0$, in which case, either (i) $\displaystyle A = 3I$ or $\displaystyle A = 5I$, or (ii) $\displaystyle (A - 5I) (A - 3I) = 0$ with $\displaystyle A - 5I \neq 0 $ and $\displaystyle A -3I \neq 0$, and in this case, I'm not sure how to use (ii) to prove anything. Obviously (b) has something to do with the second term being -8A, but apart from this I can't really think of anything.

Any help would be greatly appreciated.