Advanced book on Mathematics Olympiad

2.4 Abstract Algebra 87

263.LetAbe a square matrix whose off-diagonal entries are positive. Prove that the

rightmost eigenvalue ofAin the complex plane is real and all other eigenvalues are

strictly to its left in the complex plane.

264.Letaij,i, j = 1 , 2 ,3, be real numbers such thataij is positive fori =jand

negative fori   =j. Prove that there exist positive real numbersc 1 ,c 2 ,c 3 such that

the numbers

a 11 c 1 +a 12 c 2 +a 13 c 3 ,a 21 c 1 +a 22 c 2 +a 23 c 3 ,a 31 c 1 +a 32 c 2 +a 33 c 3

are all negative, all positive, or all zero.

265.Letx 1 ,x 2 ,...,xnbe differentiable (real-valued) functions of a single variabletthat

satisfy

dx 1

dt

=a 11 x 1 +a 12 x 2 +···+a 1 nxn,

dx 2

dt

=a 21 x 1 +a 22 x 2 +···+a 2 nxn,

···

dxn

dt

=an 1 x 1 +an 2 x 2 +···+annxn,

for some constantsaij>0. Suppose that for alli,xi(t)→0ast→∞. Are the

functionsx 1 ,x 2 ,...,xnnecessarily linearly independent?

266.For a positive integernand any real numberc, define(xk)k≥ 0 recursively byx 0 =0,

x 1 =1, and fork≥0,

xk+ 2 =

cxk+ 1 −(n−k)xk

k+ 1

Fixnand then takecto be the largest value for whichxn+ 1 =0. Findxkin terms

ofnandk,1≤k≤n.

2.4 Abstract Algebra...............................................

2.4.1 Binary Operations........................................

A binary operation∗on a setSassociates to each pair(a, b)∈S×San elementa∗b∈S.
The operation is called associative ifa∗(b∗c)=(a∗b)∗cfor alla, b, c∈S, and
commutative ifa∗b=b∗afor alla, b∈S. If there exists an elementesuch that
a∗e=e∗a =afor alla ∈S, theneis called an identity element. If an identity
exists, it is unique. In this case, if for an elementa∈Sthere existsb∈Ssuch that