Advanced book on Mathematics Olympiad

2.3 Linear Algebra 71

bkm=(− 1 )k+m   (x 1 ,x 2 ,...,xn)−^1   (x 1 ,...,xk− 1 ,xk+ 1 ,...,xn)

×Sn− 1 (x 1 ,...,xk− 1 ,xk+ 1 ,...,xn).

HereSn− 1 denotes the(n− 1 )st symmetric polynomial inn−1 variables.

222.LetAandBbe 2×2 matrices with integer entries such thatA,A+B,A+ 2 B,
A+ 3 B, andA+ 4 Bare all invertible matrices whose inverses have integer entries.
Prove thatA+ 5 Bis invertible and that its inverse has integer entries.

223.Determine the matrixAknowing that its adjoint matrix (the one used in the com-
putation of the inverse) is

A∗=

⎛

⎝

m^2 − 11 −m 1 −m

1 −mm^2 − 11 −m

1 −m 1 −mm^2 − 1

⎞

⎠,m = 1 ,− 2.

224.LetA=(aij)ijbe ann×nmatrix such that

∑n

j= 1 |aij|<1 for eachi. Prove that

In−Ais invertible.

225.Letα=nπ+ 1 ,n>2. Prove that then×nmatrix
⎛
⎜
⎜⎜
⎝

sinα sin 2α ···sinnα

sin 2α sin 4α ···sin 2nα

..

.

..

.

... ..

.

sinnαsin 2nα···sinn^2 α

⎞

⎟

⎟⎟

⎠

is invertible.

226.Assume thatAandBare invertible complexn×nmatrices such thati(A†B−B†A)
is positive semidefinite, whereX†=X
t
, the transpose conjugate ofX. Prove that
A+iBis invertible. (A matrixTis positive semidefinite if〈Tv,v〉≥0 for all
vectorsv, where〈v, w〉=vtw ̄is the complex inner product.)
We continue with problems that exploit the ring structure of the set ofn×nmatrices.
There are some special properties that matrices satisfy that do not hold in arbitrary rings.
For example, ann×nmatrixAis either a zero divisor (there exist nonzero matricesB
andCsuch thatAB=CA=On), or it is invertible. Also, if a matrix has a left (or right)
inverse, then the matrix is invertible, which means that ifAB=Inthen alsoBA=In.
A good example is a problem of I.V. Maftei that appeared in the 1982 Romanian
Mathematical Olympiad.

Example.LetA, B, Cben×nmatrices,n≥1, satisfying

ABC+AB+BC+AC+A+B+C=On.

Prove thatAandB+Ccommute if and only ifAandBCcommute.