General Matrix Question have had some problem solving the following question: (I've done part 1.):
We have two $n \times n$ matrices $A$ and $B$ and it says that $A=I-AB$.


*

*Prove that $A$ is regular and $AB=BA$. (done this one is pretty easy)

*Prove that if $B$ is symmetrical, so is $A$.

*Prove that $B^3=0$ if and only if $A=I-B+B^2$.
Thanks in advance...

For part 2, factor $A$ and obtain $A(I+B) = I$. Check the transpose and notice the symmetric parts
For part 3, the necessity direction is shown by
$$
\begin{align}
AB &= I-A\\
AB^2 &= B-AB = B-I+A\\
AB^3 &= B^2 - B + AB = B^2 - B + I - A
\end{align}
$$
if $B^3=0$ then, $B^2 - B + I - A = 0$. 
For sufficiency, 
$$\begin{align}
A &= I-B+B^2\\
I - AB &= I - B + B^2\\
-AB &= -B + B^2\\
-AB^2 &= -B^2 + B^3\\
(I-A)B^2 &= B^3\\
AB^3 &= B^3
\end{align}
$$
from this, we see that $(I-A)B^3=0$
