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Section 7.3
Matrices
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
What You Will Learn
Introduction to Matrices
Addition of Matrices
Subtraction of Matrices
Multiplying a Matrix by a Real Number
Multiplication of Matrices
7.3-2
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Matrix
A matrix is a rectangular array of
elements.
An array is a systematic arrangement
of numbers or symbols in rows and
columns.
7.3-3
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Matrix
Matrices (the plural of matrix) may be
used to display information and to
solve systems of linear equations.
The numbers in the rows and columns
of a matrix are called the elements of
the matrix.
7.3-4
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Dimensions of a Matrix
The dimensions of a matrix may be
indicated with the notation r × s, where
r is the number of rows and s is the
number of columns of a matrix.
This is a 2 × 4 matrix.
 102 93 22 35 


 82 94 23 49 
7.3-5
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Dimensions of a Matrix
A matrix that contains the same
number of rows and columns is called a
square matrix.
2×2
square matrix
 2 3 


 5 2 
7.3-6
3×3
square matrix
 4 0 2 


 4 1 3 
 6 8 9 
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Equal Matrices
Two matrices are equal if and only if
they have the same elements in the
same relative positions.
 4 0 2 
 4 0 2 

 

 4 1 3 
 4 1 3 
 6 8 9 
 6 8 9 
7.3-7
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Addition and Subtraction of
Matrices
Two matrices can only be added or
subtracted if they have the same
dimensions.
The corresponding elements of the two
matrices are either added or
subtracted.
7.3-8
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Example 4: Subtracting Matrices
Determine A – B if
 3 6 
 2 4 
A
 and B  

 8 3 
 5 1 
Solution
 3 6   2 4 
AB  


 5 1   8 3 
 3  2 6  (4)  

1
10



 5  8 1  (3)   3 2 
7.3-9
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Multiplying a Matrix by a Real
Number
A matrix may be multiplied by a real
number, a scalar, by multiplying each
entry in the matrix by the real number.
 e f
A
 h i
7.3-10
g 

j 
 3e 3f
3A  
 3h 3i
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
3g 

3 j 
Multiplication of Matrices
Multiplication of matrices is possible
only when the number of columns in
the first matrix is the same as the
number of rows of the second matrix.
For example, we can multiply a 2 × 2
matrix times a 2 × 3 matrix.
7.3-11
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Multiplication of Matrices
In general,
 a b  e f
AB  

 c d   h i
g 

j 
 ae  bh af  bi ag  bj 


 ce  dh cf  di cg  dj 
Note that Matrix Multiplication is not
commutative: A × B ≠ B × A.
7.3-12
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Example 7: Multiplying Matrices
Determine A × B if
 5 1 3 
 3 1 
A

 and B  
 4 2 
 2 8 0 
Solution
 3 1   5 1 3 
AB  


 4 2  2 8 0 
 3(5)  1(2) 3(1)  1(8) 3(3)  1(0) 


 4(5)  2(2) 4(1)  2(8) 4(3)  2(0) 
7.3-13
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Multiplicative Identity Matrix
Square matrices have a
multiplicative identity matrix.
Note that in any multiplicative identity
matrix, 1’s go diagonally from top left
to bottom right and all other elements
in the matrix are 0’s.
 1 0 0 


 1 0 
I

0
1
0


I

 0 0 1 
 0 1 
7.3-14
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Multiplicative Identity Matrix
For any square matrix A,
A × I = I × A = A.
 4 3 
If A  
 then,
 2 1 
 4 3  1 0   4 3 
A I  
 

A
 2 1  0 1   2 1 
7.3-15
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Example 9: A Manufacturing
Application
The Fancy Frock Company
manufactures three types of women’s
outfits: a dress, a two-piece suit (skirt
and jacket), and a three-piece suit
(skirt, jacket, and a vest). On a
particular day, the firm produces 20
dresses, 30 two-piece suits, and 50
three-piece suits. Each dress requires
4 units of material and 1 hour of work
7.3-16
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 9: A Manufacturing
Application
to produce, each two-piece suit
requires 5 units of material and 2
hours of work to produce, and each
three-piece suit requires 6 units of
material and 3 hours to produce. Use
matrix multiplication to determine the
total number of units of material and
the total number of hours needed for
that day’s production.
7.3-17
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 9: A Manufacturing
Application
Solution
Two- ThreeDress piece piece
A   20 30 50 
Material Hours
 4 1 


B 5 2 
 6 3 
7.3-18
Dress
Two-piece
Three-piece
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 9: A Manufacturing
Application
Solution
The product of A and B, will give the
total number of units of material and
the total number of hours of work
needed for that day’s production.
 4 1 




A  B   20 30 50  5 2


 6 3 
7.3-19
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 9: A Manufacturing
Application
 4 1 
Solution




A  B   20 30 50  5 2


 6 3 
  20(4)  30(5)  50(6) 20(1)  30(2)  50(3) 


  530 230 
A total of 530 units of material and a
total of 230 hours of work are needed
that day.
7.3-20
Copyright 2013, 2010, 2007, Pearson, Education, Inc.