Question

**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.]

Answer #1

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.

A triangular matrix is called unit triangular if
it
is square and every main diagonal element is a 1.
(a) If A can be carried by the gaussian algorithm
to
row-echelon form using no row interchanges,
show that A = LU where L is unit lower
triangular and U is upper triangular.
(b) Show that the factorization in (a) is
unique.

Show that if A is an (n × n) upper triangular matrix or lower
triangular matrix, its eigenvalues are the entries on its main
diagonal. (You may limit yourself to the (3 × 3) case.)

Calculate each limit below, if it exists. If a limit does not
exist, explain why. Show all work.
\lim _{x\to 3}\left(\frac{x-3}{\sqrt{2x+3}-\sqrt{3x}}\right)
\lim _{x\to -\infty }\left(\frac{\sqrt{x^2+3x}}{3x+1}\right)

n x n matrix A, where n >= 3. Select 3 statements from the
invertible matrix theorem below and show that all 3 statements are
true or false. Make sure to clearly explain and justify your
work.
A=
-1 , 7, 9
7 , 7, 10
-3, -6, -4
The equation A has only the trivial solution.
5. The columns of A form a linearly independent set.
6. The linear transformation x → Ax is one-to-one.
7. The equation Ax...

Which of the following are NECESSARY CONDITIONS for an n x n
matrix A to be diagonalizable?
i) A has n distinct eigenvalues
ii) A has n linearly independent eigenvectors
iii) The algebraic multiplicity of each eigenvalue equals its
geometric multiplicity
iv) A is invertible
v) The columns of A are linearly independent
NOTE: The answer is more than 1 option.

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 B = [ aij ] 20×17 be a matrix with real entries. Let x be in
R 17 , c be in R 20, and 0 be the vector with all zero entries.
Show that each of the following statements implies the other.
(a) Bx = 0 has only the trivial solution x = 0 n R 17, then (b)
If Bx = c has a solution for some vector c in R 20, then the
solution is unique.

Let n be a positive integer and p and
r two real numbers in the interval (0,1). Two random
variables X and Y are defined on a the same
sample space. All we know about them is that
X∼Geom(p) and
Y∼Bin(n,r). (In particular, we do not
know whether X and Y are independent.) For each
expectation below, decide whether it can be calculated with this
information, and if it can, give its value (in terms of p,
n, and r)....

Suppose that x has a binomial distribution with
n = 200 and p = .4.
1. Show that the normal approximation to the binomial can
appropriately be used to calculate probabilities for
Make continuity corrections for each of the
following, and then use the normal approximation to the binomial to
find each probability:
P(x = 80)
P(x ≤ 95)
P(x < 65)
P(x ≥ 100)
P(x > 100)

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