University of Phoenix Axia College
MAT 205: FINITE MATH
MAT 205
WEEK 4
CUMULATIVE TEST 2
(CT2)
INSTRUCTOR: CARY SOHL
WEEK 4 CUMULATIVE TEST
CT2
2/6/2016
1
University of Phoenix Axia College
1.
MAT 205: FINITE MATH
What is the size of the matrix below? What is its additive inverse?
1 2 3
 1 2 3


Show your work here.
The size of the matrix is 2 x 3
1 2 3
The additive inverse is 

1 2 3 

2.
The identity matrix I is defined below. Using the definition of matrix
multiplication, and writing out the element multiplications, prove for any
matrix A, also defined below, that AI=IA=A.
1 0 0 
a b


I= 0 1 0  , A=  d e
0 0 1 
 g h
c
f 
i 
Show your work here.
INSTRUCTOR: CARY SOHL
WEEK 4 CUMULATIVE TEST
CT2
2/6/2016
2
University of Phoenix Axia College
a b

AI  d e

g h
FINITE MATH MTH 205
c 1 0 0 


f 0 1 0 
i 

0 0 1 

a(1)  b(0)  c(0) a(0)  b(1)  c(0) a(0)  b(0)  c(1) 


AI =  d(1)  e(0)  f (0) d(0)  e(1)  f (0) d(0)  e(0)  f (1)

g(1)  h(0)  i(0) g(0)  h(1)  i(0) g(0)  h(0)  i(1) 

a b

 d e

g h
c 

f  A

i 

1 0 0a b


IA  0 1 0d e

0 0 1

g h
c 

f 
i 

1(a)  0(b)  0(c) 0(a) 1(b)  0(c) 0(a)  0(b) 1(c) 


 1(d)  0(e)  0( f ) 0(d) 1(e)  0( f ) 0(d)  0(e) 1( f )

1(g)  0(h)  0(i) 0(g) 1(h)  0(i) 0(g)  0(h) 1(i) 

a b

 d e

g h
c 

f  A
i 


3.
Two generalized square matrices A, B, are shown below. Is matrix
multiplication for square matrices commutative? Prove your answer using
the definition of matrix multiplication. Show a 2x2 example that supports
your proof. Hint; consider one element of the multiplication.
INSTRUCTOR: CARY SOHL
WEEK 4 CUMULATIVE TEST
CT2
2/6/2016
3
University of Phoenix Axia College
 a11 a12
A=  a21 a22

 a31 a32
FINITE MATH MTH 205
a13 
 b11 b12

a23  , B= b21 b22
b31 b32
a33 
b13 
b23 
b33 
Show your work here.
The first row first column element of AB is:
a11b11 + a12b212 + a13b31
The first row first column element of BA is:
b11a11 + b12a21 + b13a31
Clearly the two elements are not equal in a general sense, although there
may be particular matrices for which the two elements would be equal.
2x2 Example:
1 2 4 3 1(4)  2(2) 1(3)  2(1)  8 5 


 
 

3 4 2 1  3(4)  4(2) 3(3)  4(1)  20 13
4 31 2  4(1)  3(3) 4(2)  3(4)  13 20 


 
 

2 1 3 4  2(1) 1(3) 2(2) 1(4)  5 8 
8 5  13 20 

 

20 13 5 8 

MAT 205
WEEK 1 CHECKPOINT
2/6/2016
4
University of Phoenix Axia College
4.
FINITE MATH MTH 205
Find the inverse of A using text to describe your row operations, and
equation editor to show the row operation results. When you get the
inverse, A-1, get its inverse, (A-1)-1 and show that it is the original matrix
below.
 1 2 
A= 

3 4
Show your work here.
Inverse of A:
1 2 1 0 


 3 4 0 1 
Add 3 times Row 1 to Row 2 :
1 2 1 0 


0 2 3 1 
Multiply Row 1 by -1:
1 2 1 0 


0 2 3 1 
Multiply Row 2 by -1/2 :
1 2 1
0 


0 1 3/2 1/2 
Add - 2 times Row 2 to Row 1 :
1 0
2
1 


0 1 3/2 1/2 
 2
1 
A 1  

3/2 1/2 

INSTRUCTOR: CARY SOHL
WEEK 4 CUMULATIVE TEST
CT2
2/6/2016
5
University of Phoenix Axia College
FINITE MATH MTH 205
Inverse of Inverse of A:
 2
1
1 0 


3/2 1/2 0 1 
Multiply Row 1 by 1/2 :
 1
1/2 1/2 0 


3/2 1/2 0 1 
Multiply Row 2 by 2 :
1 1/2 1/2 0 


3 1 0 2 
Add three times Row 1 to Row 2 :
1 1/2 1/2 0 


0 1/2 3/2 2 
Add -1 times Row 2 to Row 1 :
1 0
1 2 


0 1/2 3/2 2 
Multiply Row 2 by 2 :
1 0 1 2 


0 1 3 4 
A 
1 1
1 2 
 
 A
 3 4 

INSTRUCTOR: CARY SOHL
WEEK 4 CUMULATIVE TEST
CT2
2/6/2016
6
University of Phoenix Axia College
5.
FINITE MATH MTH 205
Does a 3x3 matrix with 1 row with elements all equal to 0 have an
inverse? Explain your answer using the details of the augmented matrix
approach to getting an inverse.
Show your work here.
1 2 3 1 0 0 


0 0 0 0 1 0 

4 5 6 0 0 1 

1 2 3 1 0 0 


0 0 0 0 1 0 

4 5 6 0 0 1 

 6.
In the equations below, x1 and x2 are variables, and all the other quantities
are constants. Put the equations in matrix form, and use matrix inversion
to solve for x1 and x2.
a11x1+a12x2=c1
a21x1+a22x2=c2
Show your work here.
See next page.
MAT 205
WEEK 1 CHECKPOINT
2/6/2016
7
University of Phoenix Axia College
FINITE MATH MTH 205
a11 a12 x1  c1 

   
a21 a22 x 2  c 2 
x1  a11 a12 1 c1 
  
  
x 2  a21 a22  c 2 
 a22 a12 

x1  
a21 a11 c1 
 
 
a11 a12 c 2 
x 2 
a21 a22
x1 
 
x 2 
 a22 a12 


a21 a11  c1 
 
a11a22  a21a12 c 2 

a22
x1  a11a22  a21a12
  
a21
x 2  

a11a22  a21a12

a12
a11a22  a21a12 c1 
 
a11
c 2 

a11a22  a21a12 


a22
a12
c

c
1
2

x1  a11a22  a21a12
a11a22  a21a12 



 
a21
a11
x 2  
c1 
c 

a11a22  a21a12 2 
a11a22  a21a12


7.
Write the solutions that can be read from the following simplex
maximization tableau.
INSTRUCTOR: CARY SOHL
WEEK 4 CUMULATIVE TEST
CT2
2/6/2016
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University of Phoenix Axia College
x1 x2
0 2
0 3

7 4

 0 4
FINITE MATH MTH 205
x3
0
1
0
0
s1
5
0
0
0
s2
2
1
3
4
s3
2
2
5
3
z
0 15 
0 2 
0 35 

2 40 
Show your work here.
2z = 40  z = 20
7x1 = 35  x1 = 5
x3 = 2
5s1 = 15  s1 = 3
x2 = s2 = s3 = 0
8.
Use slack variables as needed, write the initial simplex tableau, then find
the solution and the maximum value.
Maximize
Subject to:
With:
z=2x1+3x2
3x1+5x2  29
2x1+x2  10
x1  0 and x2  0
Show your work here.
See next page.
INSTRUCTOR: CARY SOHL
WEEK 4 CUMULATIVE TEST
CT2
2/6/2016
9
University of Phoenix Axia College
FINITE MATH MTH 205
 3 5 1 0 0 29 


 2 1 0 1 0 10 

2 3 0 0 1 0 

Pivot about the " 5" in the first row.
Multiply Row 1 by 1/5
3/5 1 1/5 0 0 29 /5 


1
0 1 0 10 
 2

0 
2 3 0 0 1

Subtract the new Row 1 from Row 2
3/5 1 1/5 0 0 29 /5 


7 /5 0 1/5 1 0 21/5 

0
0 1
0 
2 3

Add 3 times the new Row 1 to Row 3
 3/5 1 1/5 0 0 29 /5 


7 /5 0 1/5 1 0 21/5 

1/5 0 3/5 0 1 87 /5 

Add - 3/7 times Row 2 to Row 1
 0
1 10 /35 3/7 0
4 


1
0 21/5 
7 /5 0 1/5

0
1 87 /5 
1/5 0 3/5

Multiply Row 2 by 5/7
 0
1 10 /35 3/7 0
4 


0 1/7 5 /7 0
3 
 1

0
1 87 /5 
1/5 0 3/5

INSTRUCTOR: CARY SOHL
Add 1/5 times Row 2 to Row 3
0 1 10 /35 3/7 0
1 0
0 0
1/7
4 /7
5 /7
1/5
WEEK 4 CUMULATIVE TEST
CT2
2/6/2016
10
4
0 3
1 18
University of Phoenix Axia College
9.
FINITE MATH MTH 205
State the dual problem for this linear programming problem.
Maximize
Subject to:
With:
z=8x1+3x2+x3
7x1+6x2+8x3  18
4x1+5x2+10x3  20
x1  0 , x2  0 , and x3  0
Show your work here.
Write the augmented matrix for the maximization problem:
7 6 8 18 


4 5 10 20 

8 3 1 0 

Find the transpose of that matrix:

7 4 8 


6 5 3
8 10 1 


18 20 0 
Write the minimization problem from the transposed matrix:

Minimize z = 18y1 + 20y2
Subject to: 7y1 + 4y2 ≥ 8
6y1 + 5y2 ≥ 3
8y1 + 10y2 ≥ 1
With:
y1 ≥ 0, y2 ≥ 0
10.
Use the simplex method to solve.
Find:
y1  0, y2  0 such that
3y1+y2  12
y1+4y2  16
INSTRUCTOR: CARY SOHL
WEEK 4 CUMULATIVE TEST
CT2
2/6/2016
11
University of Phoenix Axia College
FINITE MATH MTH 205
and w=2y1+y2 is minimized.
Show your work here.
Form the augmented matrix:
3 1 12 


1 4 16 

2 1 0 

Form the transpose of that matrix:

3 1 2 


1 4 1 

12 16 0 

Write the corresponding maximization problem:

Maximize w = 12x1 + 16x2
Subject to: 3x1 + x2 ≤ 2
x1 + 4x2 ≤ 1
x1 ≥ 0, x2 ≥ 0
The augmented matrix for the maximization problem is:
INSTRUCTOR: CARY SOHL
WEEK 4 CUMULATIVE TEST
CT2
2/6/2016
12
University of Phoenix Axia College
FINITE MATH MTH 205
 3
1 1 0 0 2 


4 0 1 0 1 
 1

12 16 0 0 1 0 

Pivot about the 4 in the second row.
Multiply Row 2 by 1/4
 3
1 1 0

1 0 1/4
1/4

12 16 0 0
:
2 

0 1/4 
1 0 

0
Subtract Row 2 from Row 1 :
11/4 0 1 1/4 0 7 /4 


1 0 1/4 0 1/4 
1/4

0
1 0 
12 16 0

Add 16 times Row 2 to Row 3 :
11/4 0 1 1/4 0 7 /4 


1/4 1 0 1/4 0 1/4 

4
1 4 
 8 0 0

Pivot about the 11/4 in the first row
Multiply Row 1 by 4/11 :
:
 1 0 4 /11 1/11 0 7 /11


0
1/4 0 1/4 
1/4 1

0
4
1
4 
8 0


INSTRUCTOR: CARY SOHL
WEEK 4 CUMULATIVE TEST
CT2
2/6/2016
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Add
 1

 0

8
FINITE MATH MTH 205
-1/4 times Row 1 to Row 2 :
0 4 /11 1/11 0 7 /11

1 1/11 6 /22 0 1/11 
0
0
4
1
4 

Add 8 times Row 1 to Row 3 :
1 0 4 /11 1/11 0 7 /11 


0 1 1/11 3/11 0 1/11 

0 0 32 /11 36 /11 1 100 /11

The solutions come from the columns for the slack variables:

W = 100/11, with y1 = 32/11, y2 = 36/11
INSTRUCTOR: CARY SOHL
WEEK 4 CUMULATIVE TEST
CT2
2/6/2016
14