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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.
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
Full Text: DOI
[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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