Showing $A+B$ is invertible?

$2I_n=A^{15}+B^{15}=(A+B)(A^{14}-A^{13}B+......-AB^{13}+B^{14})$

Since $A^3 = B^5 = I_n$, eigenvalue of $A$ is the root of $x^3-1=0$ and eigenvalue of $B$ is the root of $x^5-1=0$. Since both equations have only simple roots, $A$ and $B$ are diagonalizable over $\Bbb{C}$. In addition $AB=BA$, $A$ and $B$ are simultaneously diagonalizable. Thus the eigenvalue of $A+B$ is $e^{2ki\pi/3}+e^{2ji\pi/5}$, where $0\leqslant k \leqslant 2, \:0\leqslant j \leqslant 4$. But none of them is zero. So $A+B$ is invertible.