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Math Help - Simple Invertibility Question

  1. #1
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    Simple Invertibility Question

    Determine whether T is invertible, and compute T^-1 if it exists:

    T: M_{2x2}(R) \rightarrow R^4 defined by

    T(A) = (tr(A), tr(A^T), tr(EA), tr(AE)), where E = \left({\begin{array}{cc} 0 & 1  \\    1 & 0   \end{array}} \right)

    so far I have:
    tr(A)= (1, 0, 0, 1) \\ tr(A^T) = (1, 0, 0, 1) \\ tr(EA) = (0, 1, 0, 1) \\ tr(AE) = (0, 1, 1, 0)

    = \left({\begin{array}{cccc} 1 & 1 & 0 & 0  \\   0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \\ 1 & 1 & 1 & 0  \end{array}} \right)

    is this correct, or did I leave something out? I'm thinking that since the Rank is 3 and not 4, that this transformation is not invertible.
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  2. #2
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    Quote Originally Posted by lllll View Post
    Determine whether T is invertible, and compute T^-1 if it exists:

    T: M_{2x2}(R) \rightarrow R^4 defined by

    T(A) = (tr(A), tr(A^T), tr(EA), tr(AE)), where E = \left({\begin{array}{cc} 0 & 1  \\    1 & 0   \end{array}} \right)
    Since \text{tr}(A^{\textsc t}) = \text{tr}(A) and \text{tr}(EA) = \text{tr}(AE), the map T has rank 2, so certainly cannot be invertible.
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