Question

Let A be an n×n matrix. If there exists k > n such that A^k
=0,then

(a) prove that In − A is nonsingular, where In is the n × n
identity matrix;

(b) show that there exists r ≤ n such that A^r= 0.

Answer #1

Show that for each nonsingular n x n matrix A there
exists a permutation matrix P such that P A has an LR
decomposition.
[note: A factorization of a matrix A into a product A=LR of a
lower (left) triangular matrix L and an upper (right) triangular
matrix R is called an LR decomposition of A.]

Let A be an n×n nonsingular matrix. Denote by adj(A) the
adjugate matrix
of A. Prove:
1) det(adj(A)) = (det(A))
2) adj(adj(A)) = (det(A))n−2A

Let x1, x2, ..., xk be linearly independent vectors in R n and
let A be a nonsingular n × n matrix. Define yi = Axi for i = 1, 2,
..., k. Show that y1, y2, ..., yk are linearly independent.

Exercise1.2.1: Prove that if t > 0 (t∈R),
then there exists an n∈N such that 1/n^2 < t.
Exercise1.2.2: Prove that if t ≥ 0(t∈R), then
there exists an n∈N such that n−1≤ t < n.
Exercise1.2.8: Show that for any two real
numbers x and y such that x < y, there exists an irrational
number s such that x < s < y. Hint: Apply the density of Q to
x/(√2) and y/(√2).

Let A be an m × n matrix, and Q be an n × n invertible
matrix.
(1) Show that R(A) = R(AQ), and use this result to show that
rank(AQ) = rank(A);
(2) Show that rank(AQ) = rank(A).

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

Let R*= R\ {0} be the set of nonzero real
numbers. Let
G= {2x2 matrix: row 1(a b) row 2 (0 a) | a in
R*, b in R}
(a) Prove that G is a subgroup of GL(2,R)
(b) Prove that G is Abelian

Let A be an n x M matrix and let T(x) =A(x). Prove that T: R^m
R^n is a linear transformation

Let Zn = {0, 1, 2, . . . , n − 1}, let · represent
multiplication (mod n), and let a ∈ Zn. Prove that there exists b ∈
Zn such that a · b = 1 if and only if gcd(a, n) = 1.

Let P be a 2 x 2 stochastic matrix. Prove that there exists a 2
x 1 state matrix X with nonnegative entries such that P X = X.
Hint: First prove that there exists X. I then proved that x1 and x2
had to be the same sign to finish off the proof

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