Let M be a set of real n x n matrices such that: (1) 1 ∈ M; (2) if A, B ∈ M, then just one of AB, - AB is in M; (3) if A, B ∈ M, then either AB = BA or AB = -BA; (4) if 1 ≠ A ∈ M, then there is at least one B ∈ M such that AB = - BA. Prove that M contains at most n2 matrices.
For example, we might take the set:
I = (1 0) A = (0 1) B = (1 0) C = (0 -1) (0 1) (1 0) (0 -1) (1 0)We find A2 = B2 = -C2 = I, AB = -BA =C, AC = -CA = B, CB = -BC = A.
We show first that if A is in M, then A2 = ± I. Take any B. Then BA = kAB where k2 = 1, so BA2 = k ABA = k2 A2B = A2B. So A2 commutes with every element of M. Obviously -A2 does also. But by (2) one of A2 and -A2 must be in M and (4) then guarantees a non-commuting element unless A2 = ± I.
If there are more than n2 elements, then we can find a linear combination of elements which is zero: ∑ λAA = 0 (*). If we multiply on both sides by B we get ∑ ±λAA = 0. We could add this to (*) to get a linear combination with fewer terms provided BA = AB for at least one A in (*) and BA = -AB for at least one A in (*).
The only way to guarantee a + sign is if A = I. But we can do that by multiplying by A, where A appears in the combination. The A term then becomes an I term. Guaranteeing a - sign is then easy. There must be at least one other term (apart from I). Suppose it is A. Then choose B such that AB = - BA.
So if there are more than n2 terms we can find a linear relation which we can shorten without limit. Contradiction. So there are at most n2 terms.
53rd Putnam 1992
© John Scholes
7 Jan 2001