MATHEMATICS 201-105-RE
Linear Algebra
Martin Huard
Fall 2015
I - Matrices
 2  3 5
2 7  . Find a23 , a12 and a31 .
1. Let A   1
  3 0 
2. Consider the following matrices.
 2 3 5 
2 0 0
A   0 7 7 
B  3 1 0
 0 0 1 
0 3 2
 3  2 4
1 0 0
F  2 0 3
E 

0
0
1


 4
3 5
3 0 0
C 2 1 0
0 1 0
3
G   1
 0 
1 0
D0 1
0 0
1 0
H 

0 1 
Determine which of these matrices (if any) are
a) lower triangular
b) upper triangular
c) square
d) diagonal
e) column
f) row
g) symmetric
3. Consider the following matrices.
1 3
2 3
3 4 2
A 4 2  B
C 

4 2
2 4 3


 2 1
Compute the following (where possible).
a) D  E
b) D  E
e) 2D  4E
f) 3B  C
i) tr( D)
j) tr(C )
5 1 2 
D  2 0 4
1 3 2
c) 3C
g)  2(2E  4D)
k) tr(2D  5E)
4. Using the matrices defined in 3, compute the following.
a) 2BT
e) AB
i) C(DE)
m)  DE   DT ET
T
2 1 0
E  5 2 5
2 1 1
d)  2A
h) E  E
l) tr(5E )
b) C T  A
f) BA
j) C T B
c) 2ET  3D
g) AC
k) BD
d) 2AT  3C 
h CA
l) tr( B  AT CT )
n) AAT
o) D 2
p) D3
T
Math 105
I - Matrices
a b 
2
3
n
5. Consider the matrix A  
 . Evaluate A , A , A .
0
a


6. Find x and y such that
 x 3  2  5 
  6 y   4     3

   
7. Solve for A.
 2 2
 3 5
a) 
 4A  


3 1 
 2 7 
 3 2  2 1  2 1
b) 2 A  3 



 4 4   5 3 0 3 
8. Prove that if A is a square matrix, then A AT is symmetric.
9. Let A be a symmetric matrix.
a) Show that A2 is symmetric.
b) Show that 3A2  2A  3I is symmetric.
10. Prove that if AT A  A , then
a) A is symmetric
b) A  A2
11. Let A and B be two matrices defined by
 cos 2 
cos sin  
A

sin 2  
cos sin 
a) Prove that A2  A and B2  B
b) Evaluate AB and BA.
and B  I  A
12. A matrix B is said to be the square root of a matrix A if B2  A .
2 2
a) Find two square roots of 
.
2 2
4 0
b) Find all square roots of 
.
0 9
Fall 2015
Martin Huard
2
Math 105
I - Matrices
13. A furniture shop makes non-painted desks, chairs and tables out of wood. The time it takes
to make an item is given in the following chart.
Sawing
Assembling
Sanding
Desk
3
2
2
Chair
3
1
1
Table
3
2
2
a) The shop has an order for 25 desks, 32 chairs and 16 tables. Determine the time
needed to complete the order of each of the workshops.
b) Knowing that the salary for each of the workers for sawing is $12.75 an hour, for
assembly $9.05 and for sanding $10.50, determine the production costs in salary for
the order.
c) Determine the cost for producing one item of each kind.
14. A clerk in a grocery store prepares coffee mixes from three different kinds of coffee: Kenyan,
Peruvian and Columbian. The quantities necessary, in kilograms, to make one kilogram of
each mix is given in the following chart.
M1
M2
M3
Kenyan
.3
.5
.4
Peruvian
.5
.2
.2
Columbian
.2
.3
.4
a) Knowing that the store sells every week 30 kg of the first mix, 20 kg of the second
mix and 50 kg of the third mix, how many kilograms of each kind of coffee must the
clerk order every week?
b) The store buys the coffee at a price of $5.85 the kilogram for the Kenyan coffee,
$5.75 for the Peruvian and $4.25 for the Columbian. Find how much it costs the store
to make one kilogram of each mix.
c) Knowing that the store makes a 120% profit, find at what price each mix must be
sold.
Fall 2015
Martin Huard
3
Math 105
I - Matrices
ANSWERS
1. 7, -3, 
2. a) B, C, H
b) A, H
e) G
f) None
7  2 2 
3. a) 7  2 1 
b)
3 4  1
6 
 2
18


d)   8  4 
e) 24
  4 2 
10
0 0 0
h)  0 0 0 
i) 3
 0 0 0 
4.
4 8
a) 

 6  4
14 9 
e)  16 16
 0
 4 
 37  45
i) 
5
 73
c) A, B, C, F, H
d) H
g) F, H
3 0 2
 9 12 6 
 3 2 9 
c)
 6 12  9




 1 2 3
6 4
 32  4 16 

 8 12
f) Undefined
g)  4 8  52
 0 20  20
10 0 
j) Undefined
k) 1
 11 13  2
 11 0 


c)   8  4 14 
d)   4  8
  2 7 
  3 1
8 
3  16 7
17  3
f) Undefined
g) 8 24 2
h) 
12 17 

8 4 7
 2  5 
1 
j)  24  20
k) Undefined
l) –34

 49
  8
0 
4 1 
b) 0 2 
0  2
 1  30 0 
 10 10 1


 16  10
m)  1
n) 10 20 6
 11  12 15 
 1
6 5
5
26
137

30 44
p)  14
 25  27 58
 a3 3a 2b 
 a 2 2ab 
 a n na n 1b 
3
n
A

A

5. A2  




2 
a3 
an 
0
0 a 
0
y9
6. x   7
2
4
7
1
4 4
 52 21 
7. a) A   5 3 
b) A  

11 8 
4 2
Fall 2015
l) 5
Martin Huard
10 
25 1

o)  6  14 12 
 9  7  6
4
Math 105
8.
9.
I - Matrices
 A A   A A   A  A  A A
a) A    AA  A A  AA  A
b)  3 A  2 A  3I   3  A   2 A  3I
T T
2 T
T T
T
T
T
T
2
T
T
T
2
2 T
T
(since A is symmetric, AT  A )
T
 3 AT AT  2 AT  3I
 3 A2  2 A  3I
(since A is symmetric, AT =A )
b) A  AT A
10. a) A  AT A
AT   AT A  AT  AT   AT A  A
T
T
 cos 4   sin 2  cos 2 
11. a) A2   3
3
cos  sin   sin  cos
 AA  A2
since by (a) A is symmetric
cos3  sin   sin 3  cos 

sin 2  cos 2   sin 4  
 cos 2   cos 2   sin 2  
cos sin   cos 2   sin 2   


2
2
2
cos sin   cos 2   sin 2  

sin

cos


sin





 cos 2 

cos sin 
cos sin  
A
sin 2  
B2   I  A  I 2  IA  AI  A2  I  A  A  A  I  A  B (since A2  A )
2
b) AB  A  I  A  AI  A2  A  A  0
BA   I  A A  IA  A2  A  A  0
1 1
 1 1
 2 0 
12. a) 
and 
b) 


 (4 different matrices).
1 1
 1 1
 0 3
13. a) 219 hours of sawing, 114 hours of assembling and 114 hours of sanding.
b) $ 5020.95
c) $77.35 for a desk, $57.80 for a chair and $77.35 for a table.
14. a) 39 kg of Kenyan, 29 kg of Peruvian and 32 kg of Columbian
b) $5.48 for a kilogram of M1, $5.35 for a kilogram of M2 and $5.19 for a kilogram of M3.
c) $12.06 for a kilogram of M1, $11.77 for a kilogram of M2 and $11.42 for a kilogram of
M3 .
Fall 2015
Martin Huard
5