MathDB

Problem 6

Part of 2014 VTRMC

Problems(1)

func. ineq. over M2(Z):det=1

Source: VTRMC 2014 P6

5/10/2021
Let SS denote the set of 22 by 22 matrices with integer entries and determinant 11, and let TT denote those matrices of SS which are congruent to the identity matrix I(mod3)I\pmod3 (so (abcd)∈T\begin{pmatrix}a&b\\c&d\end{pmatrix}\in T means that a,b,c,d∈Z,ad−bc=1,a,b,c,d\in\mathbb Z,ad-bc=1, and 33 divides b,c,a−1,d−1b,c,a-1,d-1).
(a) Let f:T→Rf:T\to\mathbb R be a function such that for every X,Y∈TX,Y\in T with Y≠IY\ne I, either f(XY)>f(X)f(XY)>f(X) or f(XY−1)>f(X)f(XY^{-1})>f(X). Show that given two finite nonempty subsets A,BA,B of TT, there are matrices a∈Aa\in A and b∈Bb\in B such that if a′∈Aa'\in A, b′∈Bb'\in B and a′b′=aba'b'=ab, then a′=aa'=a and b′=bb'=b. (b) Show that there is no f:S→Rf:S\to\mathbb R such that for every X,Y∈SX,Y\in S with Y≠±IY\ne\pm I, either f(XY)>f(X)f(XY)>f(X) or f(XY−1)>f(X)f(XY^{-1})>f(X).
functional equationFunctional inequalityinequalitiesMatriceslinear algebra