446
Relations, Functions, and Matrices
S e c t i o n 5 . 7 Matrices
Terminology
Data about many kinds of problems can often be represented using a rectangular
arrangement of values; such an arrangement is called a matrix. Thus
A = c
1
0 4
3 −6 8
d
is a matrix with two rows and three columns. The dimensions of the matrix are
the number of rows and columns; here A is a 2 × 3 matrix.
Elements of a matrix A are denoted by a ij , where i is the row number of the element in the matrix and j is the column number. In the example matrix A, a 23 = 8
because 8 is the element in row 2, column 3, of A.
example 61
Average temperatures in three different cities for each month can be neatly summarized in a 3 × 12 matrix. Here we interpret the 3 rows as the 3 cities and the
12 columns as the 12 months January–December. The average temperature in the
third city in April, a 34 , is 67.
A = £
23 26 38 47 58 71 78 77 69 55 39 33
14 21 33 38 44 57 61 59 49 38 25 21
35 46 54 67 78 86 91 94 89 75 62 51
§
example 62
In Practice 2(c) of Section 5.1, the binary relation 5(7, 9), (2, 5), (9, 9), (2, 7)6 was
defined on the set S = 52, 5, 7, 96. Although a set is unordered, we can impose
an ordering on the elements of S so that 2 is element 1 in S, 5 is element 2, and so
forth. The matrix R below represents this binary relation by a 1 entry in position
i, j if element i in set S is related to element j. Because (7, 9) says that element 3 is
related to element 4, element r 3,4 of R equals 1.
R = ≥
0 1 1 0
0 0 0 0
0 0 0 1
0 0 0 1
¥
example 63
Solutions to many problems can be obtained by solving systems of linear equations. Suppose, for example, that you are placing an order for coffee beans for your
sidewalk café. You want to order 70 pounds of beans, a mixture of Kona coffee
beans and Colombian coffee beans. You are willing to spend $1180; Kona coffee costs $24 per pound, and Colombian coffee costs $14 per pound. How many
pounds of each should you order?
Relations, Functions, and Matrices
S e c t i o n 5 . 7 Matrices
Terminology
Data about many kinds of problems can often be represented using a rectangular
arrangement of values; such an arrangement is called a matrix. Thus
A = c
1
0 4
3 −6 8
d
is a matrix with two rows and three columns. The dimensions of the matrix are
the number of rows and columns; here A is a 2 × 3 matrix.
Elements of a matrix A are denoted by a ij , where i is the row number of the element in the matrix and j is the column number. In the example matrix A, a 23 = 8
because 8 is the element in row 2, column 3, of A.
example 61
Average temperatures in three different cities for each month can be neatly summarized in a 3 × 12 matrix. Here we interpret the 3 rows as the 3 cities and the
12 columns as the 12 months January–December. The average temperature in the
third city in April, a 34 , is 67.
A = £
23 26 38 47 58 71 78 77 69 55 39 33
14 21 33 38 44 57 61 59 49 38 25 21
35 46 54 67 78 86 91 94 89 75 62 51
§
example 62
In Practice 2(c) of Section 5.1, the binary relation 5(7, 9), (2, 5), (9, 9), (2, 7)6 was
defined on the set S = 52, 5, 7, 96. Although a set is unordered, we can impose
an ordering on the elements of S so that 2 is element 1 in S, 5 is element 2, and so
forth. The matrix R below represents this binary relation by a 1 entry in position
i, j if element i in set S is related to element j. Because (7, 9) says that element 3 is
related to element 4, element r 3,4 of R equals 1.
R = ≥
0 1 1 0
0 0 0 0
0 0 0 1
0 0 0 1
¥
example 63
Solutions to many problems can be obtained by solving systems of linear equations. Suppose, for example, that you are placing an order for coffee beans for your
sidewalk café. You want to order 70 pounds of beans, a mixture of Kona coffee
beans and Colombian coffee beans. You are willing to spend $1180; Kona coffee costs $24 per pound, and Colombian coffee costs $14 per pound. How many
pounds of each should you order?