The problem below was proposed in the Mathematics Magazine (September, 1959) and discussed in The American Mathematics Monthly (vol. 70, n. 4, Apr 1963, p. 427) by R. A. Rosenbaum:
|(I)||The set of nonsingular n×n matrices such that the sum of the elements in each row of each matrix is 1 forms a group under multiplication.|
R. A. Rosenbaum points out that a generalization of that statement is of the kind that gets closer "to the heart of the matter" than the statement itself, by stripping away nonessentials and exposing the significant relationships. As he notes, the significant hypothesis is that the row-sums be constant - not necessarily 1, and suggests to consider another problem instead.
Let A = [aij ] be an m×n matrix, B = [bij ] be an n×p matrix, and
The reformulation has a
The set of all nonsingular n×n matrices (over any field) such that, for any one matrix, the sum of the elements of each row is constant (but perhaps not the same constant for different matrices) forms a group under multiplication.
If in the generalization a, b, c are all taken to be 1, then, with
The proof of the generalization is indeed straightforward and, were a, b, c to be replaced by 1, would not differ of that for the original statement.
|∑j cij||= ∑j ∑k aik bkj|
|= ∑k ∑j aik bkj|
|= ∑k aik ∑j bkj|
|= ∑k aik b|
|= b ∑k aik|
|= b × a|
|= a × b.|
Professor W. McWorter has remarked that, for square n×n matrices, having constant row-sums means exactly having an eigenvector 1 = (1 1 ... 1)T:
Thus we can write A1 = a1, B1 = b1, and C1 = c1, implying
|= b A1|
|= a×b 1.|
So, again, c = ab, as expected.
As Professor McWorter has observed, the latter derivation works just fine when vector 1 is replaced with any other vector v. More accurately, the following statement holds:
Let A = [aij ] and B = [bij ] be an n×n square matrices, and
Furthermore, similar claims for the addition of matrices and the multiplication by a scalar also hold true.
Copyright © 1996-2017 Alexander Bogomolny