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1.)[8 marks] Solve the following systems, or show that they are inconsistent.
a)
2 x 1 − 4 x 2 + 16 x 3 − 4 x 4 = 11
2 x 1 − 6 x 2 + 22 x 3 − 6 x 4 = 10
− 2 x 1 + 5 x 2 − 19 x 3 + 5 x 4 = − 8
b)
x 1 + 3 x 2 + 5 x 3 = − 24
2 x 1 + 7 x 2 + 11 x 3 = − 54
−x 1 + x 2 = − 6
2.)[4 marks] Consider the system
x 1 + 3 x 2 + 5 x 3 = 2
−x 1 − 2 x 2 − 3 x 3 = k
− 3 x 1 − 8 x 2 − 13 x 3 = − 5
For which value(s) of k is the system
i) consistent?
ii) inconsistent?
3.)[5 marks] Find an equation of the plane that passes through the points
P (− 4 , − 1 , −1), Q(− 2 , 0 , 1) and R(− 1 , − 2 , −3).
4.)[5 marks] The Nuts ’R’ Us company produces 3 types of trail mix, using peanuts
and almonds (among others). Trail mix I requires 100g of peanuts and 100g of almonds.
Trail mix II requires 100g of peanuts and 200g of almonds. Trail mix III requires
300g of peanuts and 100g of almonds. The company has 1.6kg of peanuts and 1.5kg of
almonds available, and wants to know how many packages of each mix can be produced,
using all the supplies.
(Note: 1kg = 1000g)
a) Define the necessary variables, and set up a system of equations in order to solve
the problem.
b) Knowing that (17 − 5 t, −1 + 2t, t), t ∈ R, is a general solution to the problem, find
all realistic solutions if the company wants only complete packages.
5.)[7 marks] Consider the matrices A =
, B =
[
]
, linebreak
C =
[
]
and D =
. Find (if possible):
a) (A
T
T − C c) B
− 1 C d) B(A + C
T )
6.)[8 marks] An economy consists of two industries: Steel and Coal. The produc-
tion of $1 of steel requires $0.50 of steel and $0.30 of coal. The production of $1 of
coal requires $0.40 of steel and $0.60 of coal. There is an external demand for $
of steel and $12000 of coal.
a) How much should each industry produce to satisfy the external demand?
b) What is the internal consumption?
c) Is each industry profitable? Justify your answer.
d) Is the economy productive? Justify your answer.
7.)[8 marks] Consider the matrices A =
(^) and B =
a) Find det(A).
b) Find adj(A).
c) Use a) and b) to find A
− 1 .
d) Use A
− 1 to solve the system AX = B.
8.)[6 marks] Given the system of equations
2 x 1 − 5 x 2 + 2 x 3 + 3 x 4 = 0
x 1 + 4 x 2 + 5 x 3 − 2 x 4 = 0
3 x 1 + 7 x 2 − 11 x 3 + 3 x 4 = 3
2 x 1 + 4 x 2 + 2 x 3 + 4 x 4 = 0
and knowing that det
= -1288, use Cramer’s rule to find x 3.
9.)[4 marks] Let A, B and C be 4 × 4 matrices such that det(A) = 4, det(B) = -
and det(C) = 7.
a) Find det(BA).
b) Find det(2C).
c) Find det(A
2 B
T C
− 1 ).
d) Find rank(A).
d) Express each of the columns of A not in the basis of Col(A) as a linear combination
of the basis vectors.
e) Let the columns of A be denoted by
a 1 ,
a 2 ,
a 3 ,
a 4 and
a 5. Which of the following
sets are linearly dependent, which ones are linearly independent? (Justify)
i) {
a 1 ,
a 4 }
ii) {
a 1 ,
a 3 ,
a 4 }.
iii) {
a 1 ,
a 2 ,
a 3 ,
a 4 ,
a 5 }.
15.)[7 marks] Use the simplex method to show that the following problem has no
solution. Also, find (x, y, z) that satisfy the constraints and for either P = −1816 or
P < −1816.
Minimize P = 3x − 4 y − 2 z
subject to
3 x − 4 y 6 12
x + y − 4 z 6 4
4 x − 2 y + 5 z 6 20
x > 0 , y > 0 , z > 0
16.)[7 marks] Use the simplex method to maximize P. Your solution should
include the maximum value and the corresponding feasible solution.
P = 4x + 3y − 8 z, subject to
x − 3 y + 2 z 6 3
− 2 y + z 6 1
x + y − 2 z 6 7
x > 0 , y > 0 , z > 0
17.)[7 marks] Farmer Pete has 300 acres on which he plants all his crops. Among
those crops are corn and wheat. Corn requires 5-worker days and $15 of capital for
each acre planted. Wheat requires 2-worker days and $20 of capital for each acre
planted. Corn yields $40 in revenue per acre, while wheat yields $30 in revenue per
acre. Farmer Pete estimates that he has at most $4000 of capital and 750-worker days
of labour available for the year. What planting strategy will maximize Farmer Pete’s
revenue?
Set up the linear programming problem as follows:
a) Define all the necessary variables.
b) State the objective, and identify the objective function (in terms of the variables).
c) State all the constraints (in terms of the variables).
d) Solve the problem using the graphical method.
ANSWERS
1.) a) Inconsistent b) (6,0,-6)
2.) i) k = − 1 ii) k 6 = − 1
3.) 2 y − z = − 1
4.) a) Let
x 1 = number of packagesof trail mix I
x 2 = number of packagesof trail mix II
x 3 = number of packagesof trail mix III
100 x 1 + 100 x 2 + 300 x 3 = 1600
100 x 1 + 200 x 2 + 100 x 3 = 1500
b) t=1 (12,1,1); t=2 (7,3,2); t=3 (2,5,3).
5.) a)
[
]
b)
[
]
c)
[
]
d) Undefined.
6.) a) $100000 Steel, $105000 Coal b) $92000 Steel, $93000 Coal
c) Only Steel is profitable, sum of column < 1 d) Yes, (I − C)
− 1
7.) a) -5 b)
c)
− 6
5
4
5
3
5
− 2
5
7
5
− 3
5
d)
8.) x 3 = −
9.) a) -8 b) 112 c) −
d) 4
10.) a)
68 b)
[
]
[
]
c) Yes (t=1/2)
11.) a) Yes b) Yes c) Yes d) Yes 12.) a) k = 0 b) k 6 = 0
13.) a) True b) True
14.) a) Rank = 3, Nullity = 2 b)
