Matrices

Hi I'm not sure if this is the right forum to post this, but if it is not, I sincerely apologize.

I am really stuck at these two questions,

1.) Find all 2x2 matrices A for which A^2=A by letting A =

This was the last part to this question, "The result "If ab=0 then a=0 or b=0 for real numbers does not have an equivalent result for matrices."
I don't know if this could be helpful but I really do not know how to do this question..

2.) Give one example which shows that if "A^2=O then A=O" is a false statement. (O is the null matrix!)

I got up to multiplying the matrices together and then equated it to 0..


Really appreciate your help!

if
then

so in 1) you have the four equation



So now you have to solve this system.

and as of for 2)



you must solve this system.
I gave an exemple (it is not unique) of a solution, but you should try to find it before checking.

Spoiler: