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Idempotent matrices with invertible transpose. (English) Zbl 1448.15040
Summary: We prove that if the transpose of every $$2\times 2$$ idempotent matrix over a division ring $$D$$, different from the identity matrix, is not invertible, then $$D$$ is commutative.
##### MSC:
 15B33 Matrices over special rings (quaternions, finite fields, etc.) 15A54 Matrices over function rings in one or more variables 15A09 Theory of matrix inversion and generalized inverses
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##### References:
 [1] R. N. Gupta, Nilpotent matrices with invertible transpose, Proc. Amer. Math. Soc. 24 (1970), 572-575. · Zbl 0195.32505 [2] N. Jacobson, Lectures in Abstract Algebra. Vol. II. Linear Algebra, D. Van Nostrand, Toronto, 1953. · Zbl 0053.21204
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