Question

1. Find the inverse of the 3 by 3 matrix :

[123]

[014]

[560] by elementary row operations

2. A couple would like to invest $150,000 so that they can earn $8500 in interest in one year. One investment portfolio option suggest investing their money

into two accounts. One account earns interest at 7.5% and the other earns interest at 5%. How much should be invested in each account?

3. Given the following maximum problem, set up the initial simplex tableau and circle the first pivot element. Do not solve the maximum problems

MAXIMIZE : P = 3x1+2x2+3x3

subject to constraints : -x1+2x2+2x3<=8

4x1-x2+6x3<=10

x1+2x2+4x3<=12, x1>=0, x2>=0, x3>=0

4.Consider the following tableau. Determine the first pivot element and perform all the pivot operations for the entire pivot column.

Then, classify the tableau that results as one that is either "solved" or one that has "no solution" or one that is "ready for another set of pivot operations"

p y1 y2 y3 s1 s2 RHS

0 2 2 0 -1 1 36

0 2 2 0 -1 1 14

ㅡㅡㅡㅡㅡㅡㅡㅡㅡㅡ

1 5 -2 0 6 -1 9

Answer #1

Duality Theory: Consider the following LP:
max 2x1+2x2+4x3
x1−2x2+2x3≤−1
3x1−2x2+4x3≤−3
x1,x2,x3≤0
Formulate a dual of this linear program. Select all the correct
objective function and constraints
1. min −y1−3y2
2. min −y1−3y2
3. y1+3y2≤2
4. −2y1−2y2≤2
5. 2y1+4y2≤4
6. y1,y2≤0

Given a LP model as:Minimize Z = 2X1+ 4X2+ 6X3
Subject to:
X1+2X2+ X3≥2
X1–X3≥1
X2+X3= 1
2X1+ X2≤3
X2, X3 ≥0, X1 urs
a) Find the standard form of the LP problem.
b) Find the starting tableau to solve the Primal LP problem by
using the M-Technique.

Given the following initial simplex tableau:
x y z u v w P
2 4 3 1 0 0 0 4
6 7 1 0 1 0 0 8
6 6 5 0 0 1 0 18
-8 -11 -4 0 0 0 1 0
The pivot element that would be selected if you follow the standard
convention taught in this course is the entry in row ,
column .
Now, use this pivot element and complete ONE STEP
using the simplex...

Give augmented matrix for this system. Find all solutions to
this system. Indicate all parameters.
x1-x2+x3+x4=1
2x2+3x3+4x4=2
x1-x2+2x3+3x4=3
x1=? x2=? x3=? x4=?

Consider the following linear program Max 5x1+5x2+3x3
St
x1+3x2+x3<=3
-x1+ 3x3<=2
2x1-x2 +2x3<=4
2x1+3x2-x3<=2
xi>=0 for i=1,2,3
Suppose that while solving this problem with Simplex method, you
arrive at the following table:
z
x1
x2
x3
x4
x5
x6
x7
rhs
Row0
1
0
-29/6
0
0
0
11/6
2/3
26/3
Row1
0
0
-4/3
1
0
0
1/3
-1/3
2/3
Row2
0
1
5/6
0
0
0
1/6
1/3
4/3
Row3
0
0
7/2
0
1
0
-1/2
0...

x y s t P
1 -3 1 0 0 12
1 2 0 1 0 3
-6 -4 0 0 1 0
The pivot element for the initial simplex tableau show is the
red 1. So we need to zero out the other elements of column x. What
is the formula used to zero out row 1 and column x?
Multiply Row _____by_______ and then
add the result to Row_____
What is the formula used to zero out row...

Consider the following linear programming problem
Maximize
$1 X1 + $3 X2
Subject To
X1 + X2 ≤ 4
Constraint A
X1 - X2 ≤ 1
Constraint B
X1, X2 ≥ 0
Constraint C
Note: Report two digits after the decimal point. Do NOT
use thousands-separators (,)
1 - Which of the following is the correct standard maximization
form for the above linear programming problem
Answer CorrectNot Correct
Answer CorrectNot Correct
Answer CorrectNot Correct
Answer CorrectNot Correct
Z - X1...

x1-5x2+x3+3x4=1
2x1-x2-3x3-x4=3
-3x1-3x3+7x3+5x4=k
1 ) There is exactly one real number k for which the system has
at least one solution; determine this k and describe all solutions
to the resulting system.
2 ) Do the solutions you found in the previous part form a
linear subspace of R4?
3 ) Recall that a least squares solution to the system of equations
Ax = b is a vector x minimizing the length |Ax=b| suppose that
{x1,x2,x3,x4} = {2,1,1,1}
is a...

Problem 2. (20 pts.) show that T is a linear transformation by
finding a matrix that implements the mapping. Note that x1, x2, ...
are not vectors but are entries in vectors. (a) T(x1, x2, x3, x4) =
(0, x1 + x2, x2 + x3, x3 + x4) (b) T(x1, x2, x3, x4) = 2x1 + 3x3 −
4x4 (T : R 4 → R)
Problem 3. (20 pts.) Which of the following statements are true
about the transformation matrix...

The matrix [−1320−69] has eigenvalues λ1=−1 and
λ2=−3.
Find eigenvectors corresponding to these eigenvalues. v⃗ 1= ⎡⎣⎢⎢
⎤⎦⎥⎥ and v⃗ 2= ⎡⎣⎢⎢ ⎤⎦⎥⎥
Find the solution to the linear system of differential equations
[x′1 x′2]=[−13 20−6 9][x1
x2] satisfying the initial conditions
[x1(0)x2(0)]=[6−9].
x1(t)= ______ x2(t)= _____

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