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Turnbull was the Regius Professor of Mathematics in the United College of St Salvator and St Leonard at the University of St Andrews from 1921.
He was interested in algebra, particularly invariant theory building on work of Gordan and Clebsch.
He was also interested in the history of mathematics. His major works include The Theory of Determinants, Matrices, and Invariants (1928), The Great Mathematicians (1929), Theory of Equations (1939), The Mathematical Discoveries of Newton (1945), and An Introduction to the Theory of Canonical Matrices (1945), which was cowritten with Aitken.
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Tell me about Turnbull's work on matrices and determinants
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