Question

Prove that for a square n ×n matrix A, Ax = b (1) has one and only one solution if and only if A is invertible; i.e., that there exists a matrix n ×n matrix B such that AB = I = B A.

NOTE 01: The statement or theorem is of the form P iff Q, where P is the statement “Equation (1) has a unique solution” and Q is the statement “The matrix A is invertible”. This means that you must prove two things: 1) P =⇒ Q and 2) Q =⇒ P. NOTE 02: For this problem the most mathematically sound (and ‘pleasing’) proof does not involve jumping to any of the statements in the list of non-singular equivalences, in, for example “LINEAR ALGEBRA IN A NUTSHELL” on the last page of the textbook for 22A “Introduction to Linear Algebra”, by Gilbert Strang unless you have derived that statement yourself directly from the fact that equation (1) has a unique solution. See if you can do the problem this way. If you just jump from one the statements in the list of non-singular equivalences, to another, without deriving the equivalence, you will receive 25% credit for this problem. HINT: What is the first step to one of the other non-singular equivalences, which you can take directly from equation (1)? In other words, find and prove a second non-singular equivalence such that equation (1) is essentially the definition of this second statement. Ask if you need help trying to prove the statement in Problem 01 this way

Answer #1

Prove that n is prime iff every linear equation ax ≡ b mod n,
with a ≠ 0 mod n, has a unique solution x mod n.

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...

a)Assume that you are given a matrix A = [aij ] ∈ R n×n with (1
≤ i, j ≤ n) and having the following interesting property:
ai1 + ai2 + ..... + ain = 0 for each i = 1, 2, ...., n
Based on this information, prove that rank(A) < n.
b) Let A ∈ R m×n be a matrix of rank r. Suppose there are right
hand sides b for which Ax = b has no solution,...

(2) Letn∈Z+ withn>1. Provethatif[a]n
isaunitinZn,thenforeach[b]n ∈Zn,theequation[a]n⊙x=[b]n has a unique
solution x ∈ Zn.
Note: You must find a solution to the equation and show that
this solution is unique.
(3) Let n ∈ Z+ with n > 1, and let [a]n, [b]n ∈ Zn with
[a]n ̸= [0]n. Prove that, if the equation [a]n ⊙ x = [b]n has no
solution x ∈ Zn, then [a]n must be a zero divisor.

Answer all of the questions true or false:
1.
a) If one row in an echelon form for an augmented matrix is [0 0 5
0 0]
b) A vector b is a linear combination of the columns of a matrix A
if and only if the
equation Ax=b has at least one solution.
c) The solution set of b is the set of all vectors of the form u =
+ p + vh
where vh is any solution...

(1 point) A Bernoulli differential equation is one of the
form
dydx+P(x)y=Q(x)yn (∗)
Observe that, if n=0 or 1, the Bernoulli equation is linear. For
other values of n, the substitution u=y1−n transforms the Bernoulli
equation into the linear equation
dudx+(1−n)P(x)u=(1−n)Q(x).dudx+(1−n)P(x)u=(1−n)Q(x).
Consider the initial value problem
y′=−y(1+9xy3), y(0)=−3.
(a) This differential equation can be written in the form (∗)
with
P(x)= ,
Q(x)= , and
n=.
(b) The substitution u= will transform it into the linear
equation
dudx+ u= .
(c) Using...

A Bernoulli differential equation is one of the form
dy/dx+P(x)y=Q(x)y^n (∗)
Observe that, if n=0 or 1, the Bernoulli equation is linear. For
other values of n, the substitution u=y^(1−n) transforms the
Bernoulli equation into the linear equation
du/dx+(1−n)P(x)u=(1−n)Q(x).
Consider the initial value problem xy′+y=−8xy^2, y(1)=−1.
(a) This differential equation can be written in the form (∗)
with P(x)=_____, Q(x)=_____, and n=_____.
(b) The substitution u=_____ will transform it into the linear
equation du/dx+______u=_____.
(c) Using the substitution in part...

A Bernoulli differential equation is one of the form
dxdy+P(x)y=Q(x)yn
Observe that, if n=0 or 1, the Bernoulli equation is linear. For
other values of n, the substitution u=y^(1−n) transforms the
Bernoulli equation into the linear equation
du/dx+(1−n)P(x)u=(1−n)Q(x)
Use an appropriate substitution to solve the equation
y'−(3/x)y=y^4/x^2 and find the solution that satisfies y(1)=1

Prove by mathematical induction that for all odd n ∈ N we have
8|(n2 − 1). To receive credit for this problem, you must show all
of your work with correct notation and language, write complete
sentences, explain your reasoning, and do not leave out any
details.
Further hints: write n=2s+1 and write your problem statement in
terms of P(s).

I. Solve the following problem:
For the following data:
1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 5, 6 n = 12
b) Calculate
1) the average or average
2) quartile-1
3) quartile-2 or medium
4) quartile-3
5) Draw box diagram (Box & Wisker)
II. PROBABILITY
1. Answer the questions using the following
contingency table, which collects the results of a study to 400
customers of a store where you want to analyze the payment
method.
_______B__________BC_____
A...

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