Download LINEAR ALGEBRA IMPA - 2016 Problem Set 4 Problem 31. Prove

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?
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