Transcript Section 11.0

Chapter 11
Section 11.0
Review of Matrices
Matrices
A matrix (despite the glamour of the movie) is a collection of numbers arranged in a
rectangle or an array. We use variables like A, B, C, …, [capital letter] to stand for a
matrix.
We use what are called double scripted variables with a lower case letter of the
matrix to refer to the entries in a matrix. The numbers in the subscript give the
position the variable is located in with the first number referring to the row and the
second number the column. The dimensions or order of the matrix is given in the
form (number of rows)  (number of columns). Don't multiply leave it this way!
4 3  1  a11 a12
0 7 1    a
 21 a22
2 

a13 
a23 
a11  4
a12  3
a13  1
a21  0
a22  7
a23 
1
2
We name this matrix A
This matrix has 2 rows and 3
columns, or has order or dimension
2  3 "read 2 by 3".
b11 b12 b13  5 7  3
b
  9 12 0 
b
b
 21 22 23  

b31 b32 b33  1 4 6 
What do we name this?
B
What is the entry b21?
9
What is the entry b12?
7
What is the variable for 4?
b32
What are the dimensions?
33
Types of Matrices
A matrix can have different adjectives that describe it depending on its dimensions.
A matrix is called square matrix if it has the same number of rows and columns. A
matrix is called row matrix if it has only one row (i.e. all of the entries are in a single
row). A matrix is called a column matrix if it only has one column (i.e. all of the
entries are in a single column).
Give the dimensions of each matrix below and determine if it is a square, row or
column matrix or if it does not fall in any of the categories.
1
 3

2

5
3
7
0
2

9
6 0 1 
12 3  8


 5 4 2 
3
5
 
1 2 3 4
5 6 7 8


 9 10 11 12


13 14 15 16
42
33
21
44
None
Square
column
Square
2 0 8
5
6
9
11
Square
row
column
3 11
1 4 


 4 8 0 0
1 3 0 0 


14
22
24
12
25
row
Square
None
row
None
 4 10 7 3 6
  1 3 9 2 4


Matrix Operations
Adding & Subtracting Matrices
The way that matrices are added or subtracted is to add or subtract their
corresponding entries. This means that the matrices must be of the same dimensions
or order. If they are not we say the two matrices are not the same dimensions we say
the matrices are nonconformable.
4   35
(6)  4  8  2
 3  6  5
A  B   1 9    3  5   1  3
9  (5)   4 4 
12  4  7 10  12  (7) (4)  10 5 6 
The matrices C and D are
nonconformable. They can not
be added even though they both
have 6 entries.
4 7 
C  D  12  3  5 6 1 0  2 6
Matrix C is 2  3
 9 1 
Matrix D is 1  6
Multiplication by a Scalar
We can multiply a matrix by a number (sometimes called a scalar) by multiplying
each entry in the matrix by the number. This operation can always be done. We say
it is always conformable.
 4   2(4)   8 
2A  2 3  2(3)   6
 7   2(7)   14 
1
3
6 2  1  13 (6)
B 
   1 (7)
7
9
12

 3
1
3
1
3
1
3
(2)
(9)
1
3
1
3
(1) 2 23

(12)   73 3

4 
1
3
We can begin to combine more than one operation at a time.
3C  4D  35 2  4 2 1  3(5) 3(2)  4(2) 4(1)  15 6   8 4  7 10
 4
28
3E  7 F  34  2 6  7 3  12  6 18   21
1
 7 
What you get here is
nonconformable since
the first matrix is 1  3
and the second matrix is
3  1.
Multiplying Matrices
This is not as obvious an operation as you might think!
It is not as easy as addition or subtraction that you get
with the corresponding entries!
What you do is to multiply each entry in a row on the matrix on the left with its
corresponding entry in a column of the matrix on the right and add them up.
AB = (rows of matrix A) (columns of matrix B)
Look at the example below:
5
2 1 7    2(5)  1(9)  7(6)   10  9  42   23
AB  
9 







4
5
3
4
(
5
)

5
(
9
)

3
(

6
)
20

45

18
47

   6 
 
 

 
23
31
The dimensions of the result are given by
the rows of A and columns of B.
21
The matrix A is 2  3 and the matrix B is 3  1. The number of columns for the
matrix on the right must be the same as the number of rows for the matrix on the
left or else they are nonconformable!
6 0  2
 4(6)  1(3) 4(0)  1(5) 4(2)  1(1)  27 5  7

3 5 1 



CD  4 1 
12
13
23
5 3 2 6 0  2
ED  
 3 5 1 
0
4
1



23
23
The matrix E and the matrix D are
nonconformable even though they are the
same dimensions. The columns and rows do
not match up!
3 2  5  3(5)  2(2)  19
FB  

 




1  2 2 1(5)  2(2)  1 
If you multiply a 2  2 matrix by a 2  1 matrix you get another 2  1 matrix!
Identity Matrices
A matrix with the same number of rows and columns is called square. A square
matrix with 1's down the top left to bottom right diagonal and 0's off that diagonal is
called the identity matrix. They come in different size identity matrices.
1 0
I2  

0
1


22
1 0 0
I 3  0 1 0
0 0 1
33
1
0
I4  
0

0
0 0 0
1 0 0
0 1 0

0 0 1
44
An identity matrix has the property that if you multiply it either on the right or left by
any conformable matrix you get the conformable matrix (i.e. InA = A and AIn = A).
The matrix In for matrices acts like the number 1 for numbers.
1 0 4 3 4  0 3  0  4 3
0 1 1 7   0  1 0  7  1 7


 
 

1 0  a11 a12   a11 a12 



0 1 a
a
a
a

  21 22   21 22 
1 0 4 4  0 4
0 1 6  0  6  6

  
  
1 0  x   x  0   x 
0 1   y   0  y    y 

  
  
Representing Matrix Multiplication
A movie theatre has two prices for movie
admission, one for children and one for adult
(people over 12). It also charges one rate for a
matinee and another for an evening movie given
in the table to the right.
Adult
Child
Matinee
$4
$2
Evening
$8
$3
Find the cost of taking 2 adults and
4 children to a matinee movie.
Find the cost of taking 2 adults and
4 children to an evening movie.
2·4 + 4·2 = 8 + 8 =16
2·8 + 4·3 = 16 + 12 =28
A matrix is a rectangular array of numbers notice what we get if we multiply the two
matrices below together.
4 2 2 2  4  4  2  8  8  16 
8 3 4   2  8  4  3   16  12  28

  
 
  
Cost of Matinee Movie
Cost of Evening Movie
In other words the matrix multiplication combines all of these calculations into one.
This enables you to represent many different calculations at once.