Matrix multiplication proof

Question

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?

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 $x=2$.
Concering 2.: $B$ might be a scaled version of $A=\begin{pmatrix}0& 1\\0 &0\end{pmatrix}$, so no.

Prasad G · Accepted Answer · 2012-04-07 11:37:58Z

up vote 1 down vote accepted

For all $x \ge 3$, The equation $(A+B)^x=0$ is true because for $x =3$, it is true.

edited Apr 7 '12 at 11:37

user21436

answered Apr 6 '12 at 11:55

Prasad G
649210

1 Answer