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On Determinants whose elements are Determinants

Published online by Cambridge University Press:  20 January 2009

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The present paper is concerned with determinants whose elements are themselves determinants. The best-known determinants of this kind are those whose elements are minors of a given determinant; these are called the “adjugate” and the “compounds” of the given determinant. Determinants whose elements are themselves determinants also occur frequently in Muir's theory of Extensionals.

The determinants considered in the present paper are of a somewhat more general type than adjugates, compounds, and extensionals; the principal result obtained is “Theorem A” (at the end of § 2), which relates to determinants whose elements are formed from any number of arrays. It is shown in § 3 that many other formulae, both new and old, may be obtained by specialising the arrays in “Theorem A.”

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 1917

References

* It is best to assign to Pn a definite sign, namely, the sign which it would have as an algebraic complement in a determinant formed by superadding a new row to the array Ps.

* Liouville's Journal 16 (1854), p. 145Google Scholar. It was rediscovered by Tanner, Lloyd, Educ. Times Rep. 28 (1877), p. 41.Google Scholar