Question

Let n be a positive integer and let U be a finite subset of Mn×n(C) which is closed under multiplication of matrices. Show that there exists a matrix A in U satisfying tr(A) ∈ {1,...,n}

Answer #1

Let
n be a positive integer and let S be a subset of n+1 elements of
the set {1,2,3,...,2n}.Show that
(a) There exist two elements of S that are relatively prime,
and
(b) There exist two elements of S, one of which divides the
other.

4) Let F be a finite field. Prove that there exists an integer n
≥ 1, such that n.1F = 0F . Show further that the smallest positive
integer with this property is a prime number.

Let A be a 2 × 2 matrix satisfying A^k = 0 for some positive
integer k. Show that A^2 = 0.

Let matrices A,B∈Mn×n(R). Show that if A and B are each similar
to some diagonal matrix, and also have the same eigenvectors (but
not necessarily the same eigenvalues), then AB=BA.

Let A be a square matrix, A != I, and suppose there exists a
positive integer m such that Am = I. Calculate det(I + A
+ A2+ ··· + Am-1).

Let n greater than or equal to 1 be a positive integers, and let
X1, X2,....., Xn be closed subsets
of R. Show that X1 U X2 U ... Xn
is also closed.

Show that the set GLm,n(R) of all mxn matrices with
the usual matrix addition and scalar multiplication is a finite
dimensional vector space
with dim GLm,n(R) = mn.
Show that if V and W be finite dimensional vector spaces with
dim V = m and dim W = n, B a basis for V and C a basis for W
then
hom(V,W)-----MatB--->C(-)-------->
GLm,n(R) is a bijective linear transformation. Hence or
otherwise, obtain dim hom(V,W).
Thank you!

Let λ be a positive irrational real number. If n is a positive
integer, choose by the Archimedean Property an integer k such that
kλ ≤ n < (k + 1)λ. Let φ(n) = n − kλ. Prove that the set of all
φ(n), n > 0, is dense in the interval [0, λ]. (Hint: Examine the
proof of the density of the rationals in the reals.)

Let n be a positive integer. Show that every abelian group of
order n is cyclic if and only if n is not divisible by the square
of any prime.

Let n be a positive integer, and let Hn denote the graph whose
vertex set is the set of all n-tuples with coordinates in {0, 1},
such that vertices u and v are adjacent if and only if they differ
in one position. For example, if n = 3, then (0, 0, 1) and (0, 1,
1) are adjacent, but (0, 0, 0) and (0, 1, 1) are not. Answer the
following with brief justification (formal proofs not
necessary):
a....

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