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Math Help - linear algebra, hard for me

  1. #1
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    linear algebra, hard for me

    Question:
    A is an nxn symmetric matrix (A^t=A). if the all diagonal elements of the matrix Asquared =AA are zero, prove that the matrix A must be a zero matrix(all elements of matrix A are zero's.

    been trying for days to figure out need help badly!!!
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  2. #2
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    If A = \left( {a_{ij} } \right) and A^2  = \left( {b_{ij} } \right),\,\,b_{ij}  = \sum\limits_{k = 1}^n {\left( {a_{ik} } \right)\left( {a_{kj} } \right)} .
    Because this is a symmetric matrix a_{ik}  = a_{ki} if the diagonal is zero consider the following.
    \left( {\forall i} \right)\;0 = \left( {b_{ii} } \right) = \sum\limits_{k = 1}^n {\left( {a_{ik} } \right)\left( {a_{ki} } \right)}  = \sum\limits_{k = 1}^n {\left( {a_{ik} } \right)\left( {a_{ik} } \right)}  = \sum\limits_{k = 1}^n {\left( {a_{ik} } \right)^2 } .
    But that can only happen if and only if \left( {\forall i,k} \right)\left[ {a_{ik}  = 0} \right].
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