Advanced book on Mathematics Olympiad

2.3 Linear Algebra 61

197.Letx 1 ,x 2 ,...,xn,n≥2, be distinct real numbers in the interval[− 1 , 1 ]. Prove that

1

t 1

+

1

t 2

+···+

1

tn

≥ 2 n−^2 ,

wheretk=

∏

j   =k|xj−xk|,k=^1 ,^2 ,...,n.

198.Forn≥1, prove the following identities:

Tn(x)

√

1 −x^2

=

(− 1 )n

1 · 3 · 5 ···( 2 n− 1 )

dn

dxn

( 1 −x^2 )n−

(^12)
,
Un(x)

√

1 −x^2 =

(− 1 )n(n+ 1 )

1 · 3 · 5 ···( 2 n+ 1 )

dn

dxn

( 1 −x^2 )n+

(^12)

2.3 Linear Algebra

2.3.1 Operations with Matrices..................................

Anm×nmatrix is an array withmrows andncolumns. The standard notation is
A=(aij)i,j, whereaijis the entry (element) in theith row andjth column. We denote
byInthen×nidentity matrix (for whichaij=1ifi=j, and 0 otherwise) and byOn
then×nzero matrix (for whichaij=0 for alli, j).
Given the matrixA=(aij)i,j,Atdenotes the transpose ofA, in which thei, jentry
isaji, andAdenotes the complex conjugate, whose entries are the complex conjugates
of the entries ofA. Also, trAis the trace ofA, namely the sum of the elements on the
main diagonal:a 11 +a 22 +···+ann.
We illustrate how matrix multiplication can be used to prove an identity satisfied by
the Fibonacci sequence (F 0 =0,F 1 =1,Fn+ 1 =Fn+Fn− 1 ,n≥1). The identity we
have in mind has already been discussed in the introductory chapter in the solution to
problem 24; we put it here in a new perspective.

Example.Prove that

Fm+n+ 1 =Fm+ 1 Fn+ 1 +FmFn, form, n≥ 0.

Solution.Consider the matrix

M=

(

11

10

)

.

An easy induction shows that forn≥1,

Mn=

(

Fn+ 1 Fn

Fn Fn− 1

)

.