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Find the general form of matrices $^1U,^2U,^3U \in \mathbb{R}^{3\times3}$such that $^iU$ is nonsingular and for every permutation $\pi : \left[{3} \right] \rightarrow \left[{3} \right]$ $^1U^{\pi\left( {1} \right)},^2U^{\pi\left( {2} \right)},^3U^{\pi\left( {3} \right)}$ are linearly dependent.

If we allow 2 of the matrices to be diagonal then I believe the only form is proportional to: $\left(\begin{array}{ccc} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{array}\right),\left(\begin{array}{ccc} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{array}\right),\left(\begin{array}{ccc} 0 & 1 & 1\\ 1 & 0 & 1\\ 1 & 1 & 0 \end{array}\right)$

But for general nonsingular matrices?

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  • $\begingroup$ Hmm, what does the symbol $^1U^{\pi(1)}$ mean? Is it the $\pi(1)$-th column of the matrix $^1U$? $\endgroup$ – user1551 Jul 1 '13 at 8:21
  • $\begingroup$ @user1551: Exactly $\endgroup$ – eladidan Jul 2 '13 at 16:45

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