Question

Complete all of the exercises as soon as possible:

1. Find all k for which E = (k 1 k, 0 2 k, 0 1 3k) is invertible.

2. Let *A* = (2 -1 0 3) and **x** = (1 -1).
You are given that **x** is an eigenvector of
*A*. What is the corresponding eigenvalue?

Answer #1

Let
A =
58
9
1
9
20
9
59
9
A has λ = 7 as an eigenvalue, with
corresponding eigenvector
1
5
, and λ = 6 as an eigenvalue, with corresponding
eigenvector
−1
4
. Find the solution to the system
y′1 =
58
9
y1 +
1
9
y2
y′2 =
20
9
y1 +
59
9
y2
that satisfies the initial conditions y1(0) =
0 and y2(0) = 3. What is the value of
y1(1)

1) In the interval [0,2π) find all the solutions possible (in
radians ) :
a) sin(x)= √3/2
b) √3 cot(x)= -1
c) cos ^2 (x) =-cos(x)
2)The following exercises show a method of solving an equation
of the form: sin( AxB C + ) = , for 0 ≤ x < 2π . Find ALL
solutions .
d) sin(3x) = - 1/2
e) sin(x + π/4) = - √2 /2
f) sin(x/2 - π/3) = 1/2

6. Let A = 3 −12 4 −1 0 −2 −1 5 −1 . 1 (a) Find all
eigenvalues of A5 (Note: If λ is an eigenvalue of A, then λ n is an
eigenvalue of A n for any integer n.). (b) Determine whether A is
invertible (Check if the constant term of the characteristic
polynomial χA(λ) is non-zero.). (c) If A is invertible, find (i)
A−1 using the Cayley-Hamilton theorem (ii) All the eigenvalues...

Given Eigenvalue 3, -2. Respective Eigenvector V1 = [1 1], V2=
[1 -1]. Find the matrix A

Given (k is a constant):
x + y + kz = 1
kx + y + z = 3
2kx + 4y + 4z = 3k +12
Find the values of "k" for which the system has:
1. A unique solution.
2. Infinitely many solutions.
3. No solution.
b. Plug k = −2 and find the solution for the system
c. Plug k = 0 and find the solutions for the system.
d. Find the solution for k = 0...

2. ?̇=??, ?= [3 −18 ; 2 −9]. (1) Find the eigenvalue of
multiplicity two and their corresponding (generalized) eigenvectors
?1= [3;?] and ?2= [?;0] respectively. (2) Let ?= ?^−1??.Find the
matrix B. (3) Find ???. (4) Find the general solution of ?̇ = ??.
(5) Let ?=??.Find the general solution of ?̇ = ??. (6) Find the
solution with initial values x(0) =[4; 1].

Let a = 2i -3k; b= i+j-k
1) Find a x b.
2) Find the vector projection of a and b.
3) Find the equation of the plane passing through a with normal
b.
4) Find the equation of the line passing through the points a and
b.
Please help and show your steps. Thank you in
advance.

Linear Algebra Project : Dominant Eigenvalue Computation
a. Apply the Power Method to estimate the dominant eigenvalue
and a corresponding eigenvector for the matrix A and initial vector
x0 below. Stop at k = 5. You can use 5 decimal places maximum if
you wish (using rounding). A = 8 0 12 1 −2 1 0 3 0 ; x0 = 1 0 0
(You can also choose any other 3 × 3 or 4 × 4 matrix instead of...

For the following exercises, find all exact solutions on [0,
2π)
23. sec(x)sin(x) − 2sin(x) = 0
25. 2cos^2 t + cos(t) = 1
31. 8sin^2 (x) + 6sin(x) + 1 = 0
32. 2cos(π/5 θ) = √3

In exercises 7-12, given the list of roots and multiplicities of
the characteristic equation, form a general solution. What is the
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r = -2, k = 3; r = 2, k = 1
Answer is apparently 4th order,
y=(c1+c2t+c3t^(2))e^(-2t)+c4e^(2t)

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