So I trying to find a 2 * 2 Matrix M where but where does not = I

So for n = 1 M=I,

for n = 2

for n = 3

for n = 4

These are just a few examples I found using algebraic methods however I can not find an example for n = 5, is there a possible reason for this?

Futhermore can you find a 2 * 2 matrix M where and ?

Also is there a better way to attempting these problems than working through several simultaneous equations?

Thanks