Advanced book on Mathematics Olympiad

2.3 Linear Algebra 69

218.LetAbe ann×nmatrix such thatA+At=On. Prove that

det(In+λA^2 )≥ 0 ,

for allλ∈R.

219.LetP(t)be a polynomial of even degree with real coefficients. Prove that the

functionf(X)=P(X)defined on the set ofn×nmatrices is not onto.

220.Letnbe an odd positive integer andAann×nmatrix with the property that

A^2 =OnorA^2 =In. Prove that det(A+In)≥det(A−In).

2.3.3 The Inverse of a Matrix...................................

Ann×nmatrixAis called invertible if there exists ann×nmatrixA−^1 such that
AA−^1 =A−^1 A=In. The inverse of a matrix can be found either by using the adjoint
matrix, which amounts to computing several determinants, or by performing row and
column operations. We illustrate how the latter method can be applied to a problem from
the first International Competition in Mathematics for University Students (1994).

Example.

(a) LetAbe ann×nsymmetric invertible matrix with positive real entries,n≥2.

Show thatA−^1 has at mostn^2 − 2 nentries equal to zero.

(b) How many entries are equal to zero in the inverse of then×nmatrix

A=

⎛

⎜⎜

⎜

⎜⎜

⎜⎜

⎝

1111 ··· 1

1222 ··· 2

1211 ··· 1

1212 ··· 2

..

.

..

.

..

.

..

.

... ..

.

1212 ··· ···

⎞

⎟⎟

⎟⎟

⎟

⎟⎟

⎠

?

Solution.∑ Denote byaij andbij the entries ofA, respectively,A−^1. Then we have

n

∑ni=^0 amibim=1, so for fixedmnot all thebim’s are equal to zero. Fork  =mwe have

i= 0 akibim=0, and from the positivity of theaki’s we conclude that at least onebimis

negative, and at least one is positive. Hence every column ofA−^1 contains at least two

nonzero elements. This proves part (a).

To compute the inverse of the matrix in part (b), we consider the extended matrix

(AIn), and using row operations we transform it into the matrix(InA−^1 ). We start with