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Thread: Invertibility

  1. #1
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    Invertibility

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

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

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