کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
4598404 | 1631083 | 2017 | 17 صفحه PDF | دانلود رایگان |
Let M be an mn×mnmn×mn matrix over a commutative ring R. Divide M into m×mm×m blocks. Assume that the blocks commute pairwise. Consider the following two procedures: (1) Evaluate the n×nn×n determinant formula at these blocks to obtain an m×mm×m matrix, and take the determinant again to obtain an element of R ; (2) Take the mn×mnmn×mn determinant of M. It is known that the two procedures give the same element of R. We prove that if only certain pairs of blocks of M commute, then the two procedures still give the same element of R, for a suitable definition of noncommutative determinants. We also derive from our result further collections of commutativity conditions that imply this equality of determinants, and we prove that our original condition is optimal under a particular constraint.
Journal: Linear Algebra and its Applications - Volume 512, 1 January 2017, Pages 202–218