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2.4 Matrices
A matrix is an ordered rectangular array of numbers. A matrix with
m rows and n columns has size m x n. The entry in the ith row and
jth column is denoted by aij.
Ex.
 2 4 1 0 
 6 7 8 3


 2 0 1 9 
4 Columns
3 Rows
a23  8
Size = Row x Column
=3x4
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Square matrix – same number of rows as columns.
Ex. Here is a 2 x 2 matrix:
1

1

3



4
5


Two matrices are equal if they have the same size and their
corresponding entries are equal.
Ex. Find x and y.
Corresponding
entries are equal
x

y

1
4 7


2 

  9 3 
3
 9
y + 1 = 4 and
x
=7
2
y = 3 and x = 14
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Addition and Subtraction of Matrices
If A and B are two matrices of the same size, then
1. The sum A + B is found by adding corresponding entries in
the two matrices.
2. The difference A – B is found by subtracting the
corresponding entries in B and A.
Also, we have the Commutative law: A + B = B + A and
Associative law (A + B) + C = A + (B + C) for addition.
A zero matrix is one in which all entries are zero. The zero
Matrix O has the property that A+O=O+A=A
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Ex. Given matrices A and B, find A + B and A – B.
 0 2 4
A   2 3 7 
 3 1 8 
1 3 0
B   3 6 7 
 2 9 1
 0 1 2  3 4  0   1 5 4 
 5 9 14 



A  B   2  3 3  6 7  7  

 3  2 1  9 8  1  5 10 7 
2  3 4  0   1 1 4
 0 1
 1 3 0 



A  B   2  (3) 3  6 7  7  

 3  2
1  9 8  (1)   1 8 9 
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Transpose of a Matrix
Transpose of a Matrix – If A is an m x n matrix with elements aij,
then the transpose of A is the n x m matrix AT with elements aji.
Example
.
1 2 3


A  4 5 6
7 8 9 
1 4

T
A  2 5
3 6
7

8
9
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Scalar Product – If A is a matrix and c is a real number, then the
scalar product cA is the matrix obtained by multiplying each
entry of A by c.
 0 2 4
Example. Given the matrix A   2 3 7 


 3 1 8 
find 5A.
 5(0) 5(2) 5(4)   0 10 20


5 A  5(2) 5(3) 5(7)    10 15 35 
 5(3) 5(1) 5(8)   15 5 40
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
The market share of motorcycles in the United States in 2001 follows:
Honda 27.9%, Harley-Davidson 21.9%, Yamaha 19.2%, Yamaha
19.2%, Suzuki 11%, Kawasaki 9.1%, and others 10.9%. The
corresponding figures for 2002 are 27.6%, 23.3%, 18.2%, 10.5%,
8.8%, and 11.6%, respectively. Express this information using a 2x6
matrix. What is the sum of all the elements in the first row? In the
second row? Is this expected? Which company gained the most
market share between 2001 and 2002?
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Solution:
H
H-D
Y
S
K
O
2001  27.9 21.9 19.2 11 9.1 10.9 
A
2002  27.6 23.3 18.2 10.5 8.8 11.6 
The sum of all elements in the first row is 100%. The sum of all
elements in the second row is 100%.
Harley-Davidson gained the most: 23.3-21.9 or 1.4%.
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.