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Math Help - Prove that it is impossible for a 2x2 matrix to be both nilpotent and idempotent

  1. #1
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    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.
    Thanks in advance.
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  2. #2
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    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...
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    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.
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  4. #4
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    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?
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  5. #5
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    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)...
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