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LINEAR ALGEBRA IMPA - 2016 INSTRUCTOR: EMANUEL CARNEIRO Problem Set 4 Problem 31. Prove or disprove: There exists an ε > 0 and a real matrix A such that: −1 0 100 A = 0 −1 − ε Problem 32. Let A and B be complex n × n matrices. Prove or disprove each of the following statements: (i) (ii) (iii) (iv) If If If If A and B are diagonalizable, then so is A + B. A and B are diagonalizable, then so is AB. A2 = A, then A is diagonalizable. A2 is diagonalizable, then A is diagonalizable. Problem 33. Let A and B be two complex n × n matrices. Let fB (x) := det(xI − B) be the characteristic polynomial of B. Show that the n × n matrix fB (A) is invertible if and only if A and B have no common eigenvalue. Problem 34. Consider the n × n matrix A defined by aij = n(i − 1) + j for all 1 ≤ i, j ≤ n. The intersection of any k lines and k columns of A form a submatrix of order k of A. Let ϕ(n) be the sum of the determinants of all submatrices of A. exists and is nonzero. (i) Find the real λ such that limn→∞ ϕ(n) nλ (ii) Determine the value of the limit above for such λ. Problem 35. Prove or disprove: for each natural n there exists an n × n matrix with real entries such that its determinant is zero, but if one changes any single entry one gets a matrix with non-zero determinant. Problem 36. Prove that for all n ≥ 2, the number of 2 × 2 matrices with integer entries in the set {0, 1, 2, . . . , n − 1} and determinant of the form kn + 1, for some integer k, is given by Y 1 3 n 1− 2 . p p prime p|n Date: 27 de janeiro de 2016. 2000 Mathematics Subject Classification. XX-XXX. Key words and phrases. XXX-XXX. 1 2 EMANUEL CARNEIRO Problem 37. Let A and B be real n × n matrices. Assume that there exist n + 1 distinct real numbers t1 , t2 , . . . , tn+1 such that the matrices Ci = A + ti B, are nilpotent (i.e. Cin i = 1, 2, . . . , n + 1, = 0). Show that A and B are nipotent. Problem 38. Suppose that there exist two invertible n × n matrices A and B, both different from the identity matrix I, and such that A7 = I ABA−1 = B 2 Show that there exists an integer k > 0 such that B k = I and determine the smallest k with this property. Can you actually give an example of such a pair of matrices A and B (for some n of your choice)? Problem 39. Let p > 2 be a prime and let F = Z/pZ be the field with p elements. Let S be the set of all 4 × 4 matrices with entries in F such that A2 = I. Determine the number of elements of S. Problem 40. What is the number of 2 × 2 matrices with entries in the field Z/pZ that have the additional property that both eigenvalues are also in Z/pZ? IMPA - Estrada Dona Castorina, 110, Rio de Janeiro, RJ, Brazil 22460-320 E-mail address: [email protected]