- The matrix equation X^2 = I is satisfied only when X = I or X = -I
- (A - B)(A + B) = A^2 - B^2 if and only if AB = BA
- If B = A^2 - 5A + 2I, then AB = BA
- AB = 0 if and only if BA = 0
- If A^3 - 7A^2 + 5I = 0, then A^4 = 49A^2 - 5A - 35I

a) I'm sure this is false, but I can't think of any way, off the top of my head, to come up with a matrix that will work.

b) (A - B)(A + B) = (A - B)(A) + (A - B)(B) = A^2 - BA + AB - B^2. For this to equal A^2 - B^2, you have to have -BA + AB = 0, which means AB = BA, right?

I have no idea what to do for c) and e), and I can't think how to come up with an example for d) which I also think is wrong.