# Thread: Prove that it is impossible for a 2x2 matrix to be both nilpotent and idempotent

1. ## Prove that it is impossible for a 2x2 matrix to be 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
A2=[0 0]
[0 0]
and a idempotent matrix is where
A2 =A

I know it isn't possible, I don't know how to prove it though.

2. ## Re: Prove that it is impossible for a 2x2 matrix to be both nilpotent and idempotent

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...

3. ## Re: Prove that it is impossible for a 2x2 matrix to be 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.

4. ## Re: Prove that it is impossible for a 2x2 matrix to be both nilpotent and idempotent

Again, are we assuming that you can't use the zero matrix as A? Because the zero matrix definitely still satisfies these conditions?

5. ## Re: Prove that it is impossible for a 2x2 matrix to be both nilpotent and idempotent

Sorry, yes. I didn't specify that. I think what they're looking for is a solution involving pronumerals (a, b, c, d)...