c
Math 166, Fall 2015, Robert
Williams
Ch 5- Matrices
Matrix Basics
If A is a matrix with m rows and n columns, then the order of A is denoted
by m × n. A is sometimes written as (ai,j )m×n or (ai,j ). The element in the ith
row and j th column is denoted ai,j .
Some matrices have special names that describe their order or special properties.
A few that we will talk about are:
1. A row vector or row matrix is a matrix that consists of a single row.
If the order of the matrix is 1 × n, then it is a row matrix of dimension n
2. A column vector or column matrix is a matrix that consists of a single
column. If the order of the matrix is m × 1, then it is a column matrix of
dimension m
3. A square matrix is a matrix where the number of rows is equal to the
number of columns (i.e. it has order n × n)
Example 1 Consider the 
following
 matrices:
10






h
i
−12
3 1 0
 C = 3 −1 −5 −12
 B=
A=


 2 
2 −1 4


7
What is the order of A, B, and C? What are a2,3 , b3,1 , and c1,4 ?
1
c
Math 166, Fall 2015, Robert
Williams
Two matrices are considered to be equal if they have the same order AND all
corresponding entries are equal.
Non-Example: No two of


1 2


3 4
the below matrices are equal



h
i
1 3
1



1 2 3 4
2 4
3
2
4

0

0
Example 2 Find the values of x and y such that

 

1 x 3
1 7 3

=

0 2 5
y 2 5
Matrix Operations
A number c is often referred to as a scalar to make it clear that we mean a
number and not a 1 × 1 matrix.
Scalar Multiplication: If c is a number and A is a matrix, then cA is the
matrix obtained by multiplying every entry in A by c


 
1 2
3 6

=
3
3 4
9 12
Transpose of a Matrix: If A = (ai,j ) is a m × n matrix, then the transpose
of A is the n × m matrix AT = (aj,i )

1


3

5
T

2

1

4 = 

2
6
2
3
5
4
6


c
Math 166, Fall 2015, Robert
Williams
Matrix Addition/Subtraction: If A and B are both m×n matrices, then A+
B (or A−B) is the matrix obtained by adding (or subtracting) the corresponding
entries of A and B. Addition and subtraction between matrices of different sizes
is not defined.

1

3
 
2
2
+
4
3

3
=
−2
6
0

2
2


Example 3 Consider the following matrices:

1
A=
4
3
1
2

−3


B= 0

−2


−3
Find each of the following:
1. 2A
2. A + B
3. A − 2B T
3

4


6

3
c
Math 166, Fall 2015, Robert
Williams
Example 4 A local movie store recieves shipments from two local warehouses.
The current stock of the warehouses is given in the tables below.
Warehouse A
Comedy
Horror
Drama
Action
DVD
103
213
172
154
Blu-ray
85
97
83
72
Warehouse B
Comedy
Horror
Drama
Action
DVD
178
97
104
126
Blu-ray
95
84
76
83
Make a matrix representing the available stock that the store can request from
these warehouses.
We let 0m×n , sometimes written simply as 0, denote the zero matrix- i.e. the
m × n matrix with all entries equal to zero. If A, B, and C are all m × n
matrices, then we have the following rules:
• A+0=0+A=A
• A−A=0
• A+B =B+A
• A + (B + C) = (A + B) + C
4
c
Math 166, Fall 2015, Robert
Williams
h
i
Row Vector × Column Vector: Let R = r1 r2 . . . rn be a row vector
 
c1
 
 
 c2 

and C = 
 ..  be a column vector such that R and C have an equal number of
.
 
cn
entries. Then we define multiplication of R and C as
 
c1
 

i
h
i  c2 
 h
 = r c + r c + ··· + r c
RC = r1 r2 . . . rn 
1 1
2 2
n n
 .. 
.
 
cn
h
Example 5 3
4
 
−1
i 
 
1  2 =
 
0
Example 6 A company has four employees: Bob, Sarah, Jane, and Chris. Bob
is paid $10.50 an hour and worked 32 hours last week. Sarah is paid $15 an
hour and worked 38 hours last week. Jane is paid $11.25 an hour and worked
28 hours last week. Chris is paid $13.75 an hour and worked 35 hours last week.
Use matrices to determine how much money the company paid its employees.
5
c
Math 166, Fall 2015, Robert
Williams
Matrix Multiplication: Let A be a m × p matrix and B be a p × n matrix.
Then AB is the m × n matrix where entry in the ith row and j th column of AB
is the product of the ith row of A with the j th column of B. Note that AB is
only defined when the number of columns in A is equal to the number of rows
in B.

1


Example 7 3

5

2 
 3

4 
 2
6

1 0


0 1
We let In = 
 .. ..
. .


5
=
1

0


. . . 0

.  be the n × n matrix with ones on the diagonal and
..
. .. 

0 0 ... 1
zero everywhere else. This matrix is called the n × n identity matrix. Matrix
...
multiplication obeys the following rules:
• In A = AIm = A where A is a n × m matrix
• A(BC) = (AB)C where A is n × m, B is m × p, and C is p × q
• A(B + C) = AB + AC where A is n × m and both B and C are m × p
6
c
Math 166, Fall 2015, Robert
Williams
Example 8 Let A, B, and C be the matrices shown below:




 
1 3
−1 0
1


A=
B=
C= 
4 3
3
7 2
Find the following:
• AC
• CA
• AB
• BA
7
c
Math 166, Fall 2015, Robert
Williams
Example 9 John wants to surprise Elaine with a homemade cake, but he only
has time to visit one store to gather the ingredients. He knows how to make
two different cakes. The first cake requires two sets of batter ingredients and
one set of icing ingredients, while the second cake only requires one set of batter
ingredients but two sets of icing ingredients. The first store he can go to sells
a set of the batter ingredients for $2.75 and a set of the icing ingredients for
$1.82. The second store that he can go to sells a set of the batter ingredients for
$3.15 and a set of the icing ingredients for $1.45. If John wants to minimize the
amount of money he spends, which cake should he make and from which store
should he buy the ingredients?
Inverse Matrix: The n × n matrix B is said to be the inverse of the n × n
matrix A if AB = BA = In . In this case, we write B = A−1 . Only square
matrices may have an inverse, but not all do. A matrix that does not have an
inverse is called a singular matrix.
If A is an n × n matrix, we find A−1 as follows:
• Form the augmented matrix [A|In ]
• If possible, use row operations to reduce [A|In ] to [In |B]. If it is not
possible, then A is not invertible.
• A−1 = B
8
c
Math 166, Fall 2015, Robert
Williams

−1
Example 10 Let A = 
2
−1
−1
is A ? Is B = B ?

6

0
 and B = 
−1
1
2
7


Example 11 Let A =  3 −6

11 3

13
1
0

. Is A = B −1 ? If not, what



−3. What is A−1 ?

8
We can re-write systems of equations using matrices. Suppose we have the
following system:
a1,1 x1 + a1,2 x2 + · · · + a1,n xn = b1
a2,1 x1 + a2,2 x2 + · · · + a2,n xn = b2
..
.
an,1 x1 + an,2 x2 + · · · + an,n xn = bn

 
 
a1,1 a1,2 . . . a1,n
x1
b1


 
 


 
 
 a2,1 a2,2 . . . a2,n 
 x2 
 b2 
, X =  , and B =  , then
Then if we let A = 
..
.
.
 ..



 .. 
..
.. 
 .
 .. 
.
.
.


 
 
an,1 an,2 . . . an,n
xn
bn
we can write the system of equations as AX = B. The system of equations has

a unique solution if and only if A is invertible. In this case, we get X = A−1 B.
9
c
Math 166, Fall 2015, Robert
Williams
Example 12 Solve the two systems of linear equations below
System 1
System 2
3x + 2y − z = 20
3x + 2y − z = 32
−x + 3y + z = 10
−x + 3y + z = 32
10x − 20y + 30z = 60
10x − 20y + 30z = 140
10