Chapter 5: Matrices
Section 5.1-5.3
5.1 Introduction to Matrices
A
array of numbers.
, A, also denoted by (aij )m×n or simply (aij ), is a rectangular
of A is denoted by m × n where m is the number of rows and
The
n is the number of columns.
The
in the ith row and jth column is denoted by aij .
Example 1 Determine the order of the following matrices.


1 4 2
a) A =  4 3 6 
3 π −1

2 1 0
9
0 
b) B =  23 0 3
1
1 1 − 6 −2

For the above matrices, what are a23 and b32 ?
A matrix of order 1 × n (1 row and n columns) is called a
matrix of dimension n.
matrix or a row
Example 2 The matrix
−3 0 2 5
is a row matrix of dimension 4.
A matrix of order m × 1 (m rows and 1 column) is called a
or a column matrix of dimension m.
1
matrix
Example 3 The matrix
 
4
 1 
0
is a column matrix of dimension 3.
matrix if the number of rows is equal to the
A matrix is called a
number of columns.
Example 4 The matrix
−1 3
0 2
is a square matrix of order 2 × 2.
Matrix Operations:
1.
: If c is a number and A is a matrix,
then the matrix cA is the matrix obtained by multiplying every entry in A by c.
2.
: Two matrices of the same
order can be added (or subtracted) to obtain another matrix of the same order by
adding (or subtracting) corresponding entries.
3.
: If A is an m × n matrix with elements aij
then the transpose of A is the n × m matrix AT with elements aji .
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Example 5 Consider the matrices




2 1
1 −2
A =  −3 0  and B =  5 −3 
1 4
−2 6
a) Compute 3A
b) Compute A + B
c) Compute 3A − 2B
d) Compute B T
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The
matrix of order m × n is the matrix O with m rows and n columns
all of whose entries are zero.
0 0 0
Example 6 The zero matrix of order 2 × 3 is O =
.
0 0 0
Properties of Matrix Operations: If A, B, and C are all matrices of order m × n
and O is the zero matrix of order m × n, then
(1) A + O = O + A = A
(2) A − A = O
(3) A + B = B + A (commutative property)
(4) A + (B + C) = (A + B) + C (associative property)
Two matrices are
entries are equal.
if they have the same order and all corresponding
Example 7 Find a, b, c, and d so that the following is true:
2
a+3 4
2
6
−1
6
3 d
=
9 7b − 1
c −8
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Matrices are often used to organize and work with data, not solely for solving systems of
equations.
Example 8 (Tan) The following table gives the number of shares of certain corporations
held by Olivia and Max in their stock portfolios at the beginning of September and at the
beginning of October:
September
IBM Google Boeing
Olivia 800
500
1200
Max
500
600
2000
IBM
Olivia 900
Max
700
GM
1500
800
October
Google Boeing
600
1000
500
2100
GM
1200
900
a) Write matrices A and B giving the stock portfolios of Olivia and Max at the beginning
of September and the beginning of October, respectively.
b) Find a matrix C reflecting the change in the stock portfolios of Olivia and Max between
the beginning of September and the beginning of October.
c) Find a matrix D that gives the average number of each stock portfolio for Olivia and
tha average number of each stock portfolio for Max over these two months.
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5.2 Matrix Multiplication
Multiplying a row
by a column matrix: If A = a1 a2 ... an is a row
 matrix

b1
 b2 


matrix and B =  ..  is a column matrix of the same length, then
 . 
bn


−2
Example 9 Multiply 2 3 −1 ×  4 
0
Multiplying two matrices: Given two matrices A = (aij )m×p and B = (bij )q×n , the
product AB = C = (cij ) is defined if p = q, that is, if the number of columns of A equals
the number of rows of B.
matrix.
The resulting matrix, AB, is an
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Example 10 Determine the order of the following matrices.
a) A has order 3 × 2 and B has order 2 × 2, then AB has order
.
b) A has order 2 × 2 and B has order 3 × 2, then AB has order
.
c) A has order 5 × 4 and B has order 4 × 3, then AB has order
.
The element in the ith row and jth column, cij , of the product matrix C = AB is obtained
by multiplying the ith row of A and the jth column of B.
2 1 −1
3 −2 4
7
2 3
−4 −1 0
Example 11 Let A =
Example 12 Let A =


−3 2
and let B =  1 −4 . What is AB?
5 −1
and let B =
7
−3 0 3
. What is AB?
7 −4 1
Example 13 Let A =
1 2
3 4
and let B =
a b
.
c d
a) Find AB.
b) Find BA.
Note: In general, AB does NOT equal BA. So matrix multiplication is not commutative.
Properties of Matrix Multiplication: For any matrices, A, B, and C,
(1) A(BC) = (AB)C (Associative Property)
(2) A(B + C) = AB + AC (Distributive Property)
whenever the indicated properties are defined.
We need to know how to multiply matrices by hand, but you can also use the calculator
when all the entries are numbers. Matrix multiplication can also be used in word problems. When multiplying matrices where the matrices have meaning, you need to keep
two things in mind:
1. Size: Make sure the sizes of the matrices allow you to multiply them.
2. Meaning: Label your matrices, and then make sure the labels on the columns of
the first matrix match the labels on the rows of the second matrix. Otherwise the
multiplication does not make sense.
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Example 14 My cheesecake company sells three types of cheesecake:
• Regular cheesecakes sell for $15 each
• Special cheesecakes sell for $25 each
• Super-special cheesecakes sell for $40 each
Further, our daily sales are as follows:
Regular
Special
Super-Special
Mon
13
8
6
Tues
9
7
4
What was our revenue for each day?
9
Wed
7
4
0
Thur
5
6
3
Example 15 (Tan) A university admissions committee anticipates an enrollment of 8000
students next year. To satisfy admission quotas, incoming students have been categorized
according to their gender and place of residence. The number of students in each category
is given in the table
In-state
Out-of-state
Foreign
Male
2700
800
500
Female
3000
700
300
By using data accumulated in previous years, the admissions committee has determined
that these students will elect to enter the College of Letters and Science, the College of
Fine Arts, the School of Business Administration, and the School of Engineering according
to the percentages that appear in the following table:
Male
Female
L&S
0.25
0.30
Fine Arts
0.20
0.35
Bus. Adm.
0.30
0.25
Eng.
0.25
0.10
Find the matrix AB that shows the number of in-state, out-of-state, and foreign students
expected to enter each discipline.
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5.3 Inverse of a Square Matrix
The n × n
matrix, In , is the square matrix of order n × n with
ones down the main diagonal and zeros elsewhere.
Example 16 The following are some identity matrices:

1
1 0 0
 0
1 0
I2 =
, I3 =  0 1 0 , I4 = 
 0
0 1
0 0 1
0


Example 17 Let A =
0
1
0
0
0
0
1
0

0
0 

0 
1
3 4
. Find I2 A.
2 9
Multiplication Property of the Identity Matrix: If A is a square matrix of order
n × n, then
of the n × n matrix A if
The n × n matrix B is said to be the
If the inverse matrix B exists, we write
.
Example 18 Verify that the following matrices are inverses.
A=
2 1
,B=
−1 3
3
7
1
7
− 17
2
7
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Note: We NEVER divide matrices.
Not all matrices have inverses.
Square matrices that have inverses are said to be
Square matrices that do NOT have inverses are said to be
.
.
If a matrix is not square, it CANNOT have an inverse. If a matrix is square, it may or
may not have an inverse.
For this class, we will find matrix inverses on the calculator. Just call up the matrix and
then press x−1 .
Example 19 Find the inverse of each of the following matrices if it exists.


1 1
1
a) A =  1 −1 1 
1 −2 −1


1 2 3
b) B =  2 4 6 
3 6 2
Example 20 Solve the following matrix equations for X. (Assume all matrix operations
are defined.)
a) AX = B
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b) CX + D = F X + G
c) 3X − K = N − XA
Example 21 Let A =
2 1
3 5
and X =
x
.
y
1. Find AX.
2. If B =
−2
, what must be true if AX = B?
−4
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We can also use inverse matrices to solve systems of equations. Our first step is to rewrite
the system of equations as a matrix equation:
AX = B
where A is the coefficient matrix, X is the variable matrix, and B is the constant
matrix.
We can then solve the matrix equation by multiplying (on the left) by the inverse of the
coefficient matrix, if it exists. If the inverse exists, this will give the unique solution. If
the inverse does not exist, there are either infinitely many solutions or no solutions.
Example 22 Given is a system of equations:
x + y + 2z + w
4x + 5y + 9z + w
3x + 4y + 7z + w
2x + 3y + 4z + 2w
=3
=6
=5
=7
a) Write a matrix equation that is equivalent to the system of linear equations.
b) Solve the system of equations using matrix inverses.
In practice, it is usually easier to solve a system of equations using rref because a matrix
does not always have an inverse.
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