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Math Help - Invertibility & Isomorphism

  1. #1
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    Invertibility & Isomorphism

    Let A be an n x n matrix.

    a) Suppose that A^2 = O. Prove that A is not invertible

    b) Suppose that AB = O for some nonzero n x n matrix B. Could A be invertible? Explain.
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  2. #2
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    Quote Originally Posted by leungsta View Post
    a) Suppose that A^2 = O. Prove that A is not invertible
    If AA^{-1} = I \implies A^2A^{-1} = A \implies 0 = A a contradiction.

    b) Suppose that AB = O for some nonzero n x n matrix B. Could A be invertible? Explain.
    If A is invertible then A^{-1} (AB) = A^{-1} (0) \implies B = 0 a contradiction.
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