Advanced book on Mathematics Olympiad

2.3 Linear Algebra 67 213.LetAandBbe 3×3 matrices with real elements such that detA=detB = det(A+B)=det(A−B)=0. Prove that det(x ...

68 2 Algebra Z= ( A 0 CIn ) . According to the rule of Laplace, the determinant ofZcan be computed by expanding along then×nmino ...

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 polynom ...

70 2 Algebra ⎛ ⎜⎜ ⎜⎜ ⎜⎜ ⎜ ⎝ 1111 ··· 1 1000··· 0 1222 ··· 2 0100··· 0 1211 ··· 1 0010··· 0 1212 ... 2 0001... 0 .. . .. . .. . . ...

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 ,... ...

72 2 Algebra Solution.If we addInto the left-hand side of the identity from the statement, we recognize this expression to be th ...

2.3 Linear Algebra 73 2.3.4 Systems of Linear Equations............................... A system ofmlinear equations withnunknown ...

74 2 Algebra cαx+dαy+cγ z+dγt=p, cβ x+dβy+cδz+dδt=q. We saw above that this system has a unique solution, which implies that its ...

2.3 Linear Algebra 75 ui= ∏ 3 ∏k=^1 (ai+xk) k =i(ai−ak) ,i= 1 , 2 , 3. It is not hard to check that these satisfy the system o ...

76 2 Algebra x 3 +x 5 =yx 1 ,x 4 +x 1 =yx 5 , whereyis a parameter. 233.Leta, b, c, dbe positive numbers different from 1, andx, ...

2.3 Linear Algebra 77 239.Leta 0 = 0 ,a 1 ,...,an,an+ 1 =0 be a sequence of real numbers that satisfy|ak− 1 − 2 ak+ak+ 1 |≤1 for ...

78 2 Algebra Bk=a 1 Bk+ 1 +a 2 Bk+ 2 +···+an (^2) −kBn 2 , whereaj=−ckc+kj. ComputingBk+ 1 =BkX−XBk, we obtain Bk+ 1 =a 1 Bk+ 2 ...

2.3 Linear Algebra 79 243.LetAbe then×nmatrix whosei, jentry isi+jfor alli, j= 1 , 2 ,...,n. What is the rank ofA? 244.For integ ...

80 2 Algebra The spectral mapping theorem.LetAbe ann×nmatrix with not necessarily distinct eigenvaluesλ 1 ,λ 2 ,...,λn, and letP ...

2.3 Linear Algebra 81 Solution.Choose a basis that identifiesVwithRmandWwithRn. Associate toAand Btheir matrices, denoted by the ...

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 ...

2.3 Linear Algebra 83 2.3.7 The Cayley–Hamilton and Perron–Frobenius Theorems........ We devote this section to two more advance ...

84 2 Algebra Solution.By the Cayley–Hamilton Theorem, B^2 −(trB)B+I 2 =O 2. Multiply on the left byAB−^1 to obtain AB−(trB)A+AB− ...

2.3 Linear Algebra 85 Now, there is a rather general result that states that a contractive function on a complete metric space h ...

86 2 Algebra It follows that forv, w∈K,v =w,δ(f (v), f (w)) < δ(v, w). Now,Kis closed and but is not bounded in the Hilbert n ...