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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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