The common-submatrix Laplace expansion
نویسنده
چکیده
We state and prove a classical version of the Laplace expansion theorem where all submatrices in the expansion are restricted to contain a speci ed common submatrix (CSM). The result states that (accounting for signs) this restricted expansion equals the determinant of the original matrix times the determinant of the CSM. This result (Muir [Mui60], p.132) is one of many such results contained in A Treatise on the Theory of Determinants by Muir (revised and enlarged by Metzler). Our proof, based on the same general idea used by Muir and valid for any commutative ring, is a modi cation of a proof used in Williamson [Wil86] to study the non-commuting case.
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