Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Determinants of Block Matrices with Noncommuting Blocks   Let $M$ be an $mn\times mn$ matrix over a commutative ring $R$. Divide $M$
into $m \times m$ blocks. Assume that the blocks commute pairwise. Consider the
following two procedures: (1) Evaluate the $n \times n$ determinant formula at
these blocks to obtain an $m \times m$ matrix, and take the determinant again
to obtain an element of $R$; (2) Take the $mn \times mn$ 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.

28
The categories of this paper are: alg-geom
4
The categories of this paper are: cond-mat
7
The categories of this paper are: hep-th
26
The categories of this paper are: bayes-an
1
The categories of this paper are: cs
30
The categories of this paper are: q-alg
31
The categories of this paper are: ao-sci
23
The categories of this paper are: solv-int
20
The categories of this paper are: acc-phys
22
The categories of this paper are: chao-dyn
34
The categories of this paper are: chem-ph
36
The categories of this paper are: cmp-lg
5
The categories of this paper are: gr-qc
16
The categories of this paper are: stat
35
The categories of this paper are: mtrl-th
33
The categories of this paper are: plasm-ph