# Matrix multiplication proof

I have the following details about the matrices $A,B$:

$$A_{n\times n},B_{n\times n}$$ $$A^2 = B^2 = 0$$ $$AB=BA$$

I need to find $x \in \mathbb{N}$ so $(A+B)^x=0$

What implications can I make from the given details?

1. $A,B$ doesn't have to equal $0$ so $A^2 = B^2 = 0$
2. Can I imply from the 2nd and 3rd given details that $A=B$?
Write out $(A+B)^x$ for $x=2$ and $x=3$ and simplify as far as you can. You should notice something nice. You can use the binomial theorem for these powers before simplification (because $AB=BA$). Then try $x=4$ using your result from $x=2$. – bgins Apr 6 '12 at 11:42
Concering 2.: $B$ might be a scaled version of $A=\begin{pmatrix}0& 1\\0 &0\end{pmatrix}$, so no. – draks ... Apr 6 '12 at 11:49
For all $x \ge 3$, The equation $(A+B)^x=0$ is true because for $x =3$, it is true.