Question

consider the matrix

Answer 1

(a)
Well, AA = AA so our first solution is B=A
and AI=IA so the second is C=I

(b)
Now, combining these, AA+AI = AA+IA.
We can factor out A (ensuring that we factor on the correct side..)
A(A+I) = (A+I)A
So we see that A+I is another solution.

I'm not going to go into the full induction process but it's quite easy to show that
A(A+A+I) = (A+A+I)A and carrying on..
A(kA+I) = (kA+I), k≥0.

I'll admit that I lack the proof to demonstrate that these are the only solutions..