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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!!!
If $\displaystyle A = \left( {a_{ij} } \right)$ and $\displaystyle 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 $\displaystyle a_{ik} = a_{ki}$ if the diagonal is zero consider the following.
$\displaystyle \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 $\displaystyle \left( {\forall i,k} \right)\left[ {a_{ik} = 0} \right]$.
