82 2 Algebra
249.LetA, Bbe 2×2 matrices with integer entries, such thatAB=BAand detB=1.
Prove that if det(A^3 +B^3 )=1, thenA^2 =O 2.
250.Consider then×nmatrixA=(aij)withaij=1ifj−i≡ 1 (modn)andaij= 0
otherwise. For real numbersaandbfind the eigenvalues ofaA+bAt.
251.LetAbe ann×nmatrix. Prove that there exists ann×nmatrixBsuch that
ABA=A.
252.Consider the angle formed by two half-lines in three-dimensional space. Prove that
the average of the measure of the projection of the angle onto all possible planes in
the space is equal to the angle.
253.A linear mapAon then-dimensional vector spaceV is called an involution if
A^2 =I.
(a) Prove that for every involutionAonVthere exists a basis ofVconsisting of
eigenvectors ofA.
(b) Find the maximal number of distinct pairwise commuting involutions.
254.LetAbea3×3 real matrix such that the vectorsAuanduare orthogonal for each
column vectoru∈R^3. Prove that
(a)At=−A, whereAtdenotes the transpose of the matrixA;
(b) there exists a vectorv∈R^3 such thatAu=v×ufor everyu∈R^3.
255.Denote byMn(R)the set ofn×nmatrices with real entries and letf:Mn(R)→R
be a linear function. Prove that there exists a unique matrixC∈Mn(R)such that
f (A)=tr(AC)for allA∈ Mn(R). In addition, iff (AB) =f(BA)for all
matricesAandB, prove that there existsλ∈Rsuch thatf (A)=λtrAfor any
matrixA.
256.LetUandVbe isometric linear transformations ofRn,n≥1, with the property
that‖Ux−x‖≤^12 and‖Vx−x‖≤^12 for allx∈Rnwith‖x‖=1. Prove that
‖UVU−^1 V−^1 x−x‖≤1
2
,
for allx∈Rnwith‖x‖=1.257.For ann×nmatrixAdenote byφk(A)the symmetric polynomial in the eigenvalues
λ 1 ,λ 2 ,...,λnofA,
φk(A)=∑
i 1 i 2 ···ikλi 1 λi 2 ···λik,k= 1 , 2 ,...,n.For example,φ 1 (A)is the trace andφn(A)is the determinant. Prove that for two
n×nmatricesAandB,φk(AB)=φk(BA)for allk= 1 , 2 ,...,n.