If those are the conditions for a matrix to be nilpotent and idempotent, surely the zero matrix satisfies both conditions and is therefore both nilpotent and idempotent...
The question is: "Prove that it is impossible for a 2x2 matrix to be both nilpotent and idempotent."
I am aware that a nilpotent matrix is where
A^{2}=[0 0]
[0 0]
and a idempotent matrix is where
A^{2} =A
I know it isn't possible, I don't know how to prove it though.
Thanks in advance.
If those are the conditions for a matrix to be nilpotent and idempotent, surely the zero matrix satisfies both conditions and is therefore both nilpotent and idempotent...
A matrix A is nilpotent if it is capable (potent) of becoming a nil (zero) in some power, not necessarily power 2. Also note that if A is idempotent, then every power of A equals A.