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{\bf Matja\v z Konvalinka}
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{\bf Non-Commutative Sylvester's Determinantal Identity}
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Sylvester's identity is a classical determinantal identity with a
simple linear algebra proof. We present combinatorial proofs
of several non-commutative extensions, and find a $\beta$-extension
that is both a generalization of Sylvester's identity and the
$\beta$-extension of the quantum MacMahon master theorem.

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