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Math Help - Invertibility

  1. #1
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    Invertibility

    Let A be an n \times {n} matrix
    suppose that A^2 = 0. Prove that A is not invertible.

    I know that for A to be invertible it must satisfy:
    1) T:A \rightarrow B, such that UT = I_{v} which would shows that it's 1-1.
    2) U:B \rightarrow A, such that TU = I_{w}, showing that it's onto.

    But since A^2 =0 even though A \neq 0, stipulating that the cancellation property for multiplication is not valid.

    Is this right, or am I missing a large part?
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  2. #2
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    Let us suppose that A is non zero and is invertible and arrive at a contradiction.

    If  A is invertible then there exists a non zero matrix B such that
     AB = I_n = BA where  I is the identity matrix. But this would imply
     A = AI = A(AB) = A^2 B = 0B = 0,
    a contradiction.
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