Show matrix is nilpotent
Solution 1:
As you mentioned in your another question, this is just a variant of that question.
Let $X=A-B$ and $Y=A+B$. Then \begin{align} XY&=(A-B)(A+B)=(c+1)(AB-BA),\\ YX&=(A+B)(A-B)=(c-1)(AB-BA). \end{align} Hence $XY-YX = 2(AB-BA)$ and $XY+YX = 2c(AB-BA)$, i.e. $XY+YX=c(XY-YX)$ for some $c\ne0$. So, the result from the aforementioned question shows that $(XY-YX)^n=0$. Hence $(AB-BA)^n=0$.