Transcript 4.7 Identity and Inverse Matrices

4.7 Identity and Inverse Matrices
-Identity matrices
-Inverse matrix (intro)
-An application
-Finding inverse matrices (by hand)
-Finding inverse matrices (using calculator)
A review of the Identity
• For real numbers, what is the additive
identity?
• Zero…. Why?
• Because for any real number b, 0 + b = b
• What is the multiplicative identity?
• 1 … Why?
• Because for any real number b, 1 * b = b
Identity Matrices
• The identity matrix is a square matrix
(same # of rows and columns) that, when
multiplied by another matrix, equals that
same matrix
• If A is any n x n matrix and I is the n x n
Identity matrix, then A * I = A and I*A = A
Examples
• The 2 x 2 Identity
matrix is:
1 0 
0 1 


• The 3 x 3 Identity
matrix is:
1 0 0 
0 1 0 


0 0 1 
•Notice any pattern?
•Most of the elements are 0, except those in the diagonal
from upper left to lower right, in which every element is 1!
Inverse review
• Recall that we defined the inverse of a real
number b to be a real number a such that
a and b combined to form the identity
• For example, 3 and -3 are additive
inverses since 3 + -3 = 0, the additive
identity
• Also, -2 and – ½ are multiplicative
inverses since (-2) *(- ½ ) = 1, the
multiplicative identity
Matrix Inverses
• Two n x n matrices are inverses of each other if
their product is the identity
• Not all matrices have inverses (more on this
later)
• Often we symbolize the inverse of a matrix by
writing it with an exponent of (-1)
• For example, the inverse of matrix A is A-1
• A * A-1 = I, the identity matrix.. Also A-1 *A = I
• To determine if 2 matrices are inverses, multiply
them and see if the result is the Identity matrix!
Determine whether X and Y are inverses.
Check to see if X • Y = I.
Write an
equation.
Matrix
multiplication
Now find Y • X.
Write an
equation.
Matrix
multiplication
Answer: Since X • Y = Y • X = I, X and Y are inverses.
Determine whether P and Q are inverses.
Check to see if P • Q = I.
Write an
equation.
Matrix
multiplication
Answer: Since P • Q  I, they are not inverses.
Determine whether each pair of matrices are inverses.
a.
Answer: no
b.
Answer: yes
An Application of Inverse Matrices
• You can use matrices to encode and decode a message
• In other words, matrices are useful for encrypting
information
• First, translate your message into numbers using the key
A = 1, B = 2, etc.. (perhaps 0 = space)
• Organize your message into a matrix with 2 columns and
as many rows as needed
• Multiply the matrix by a 2 x 2 encoding matrix
• To decipher the message, multiply the coded message
by a 2 x 2 decoding matrix
• The decoding matrix will be the inverse of the encoding
matrix
• Finally, you can translate the numbers back into letters
using you’re the key mentioned above
Use the table to assign a
number to each letter in the
message ALWAYS_SMILE.
Then code the message with
Code
the matrix
Convert the message to
numbers using the table.
_ 0
I
9
R 18
A 1
J 10
S 19
B 2
K 11
T 20
C 3
L 12
U 21
D 4
M 13
V 22
E 5
N 14
W23
F 6
O 15
X 24
G 7
P 16
Y 25
H 8
Q 17
Z 26
A
L
W
A
Y
S
_
S
M
I
L
E
1
12
23
1
25
19
0
19
13
9
12
5
Write the message in matrix form. Then multiply the
message matrix B by the coding matrix A.
Write an equation.
Matrix multiplication
Simplify.
Answer: The coded message is 13 | 38 | 24 | 49 | 44 |
107 | 19 | 57 | 22 | 53 | 17 | 39.
Now decode the message
13 | 38 | 24 | 49 | 44 | 107 | 19 | 57 | 22 | 53 | 17 | 39
• Decode by:
• expressing the coded message as a matrix with 2
columns
• Multiplying this matrix by the inverse of A
• The inverse of A is shown below:
 3 2 
 1 1 


Next, decode the message by multiplying the coded
matrix C by A–1.
Code
Use the table again to convert
the numbers to letters. You can
now read the message.
Answer:
_ 0
I
9
R 18
A 1
J 10
S 19
B 2
K 11
T 20
C 3
L 12
U 21
D 4
M 13
V 22
E 5
N 14
W23
F 6
O 15
X 24
G 7
P 16
Y 25
H 8
Q 17
Z 26
1
12
23
1
25
19
0
19
13
9
12
5
A
L
W
A
Y
S
_
S
M
I
L
E
a. Use the table to assign a
number to each letter in the
message FUN_MATH.
Then code the message
Code
_ 0
I
9
R 18
A 1
J 10
S 19
B 2
K 11
T 20
C 3
L 12
U 21
D 4
M 13
V 22
E 5
N 14
W23
Answer: 12 | 63 | 28 | 14 | 26 |
F 6
O 15
X 24
16 | 40 | 44
G 7
P 16
Y 25
H 8
Q 17
Z 26
with the matrix A =
Example 7-3k
Code
Use the inverse matrix shown
below to decode the message!!
 1
 2
 0

1 
6
1 
3
Answer:
_ 0
I
9
R 18
A 1
J 10
S 19
B 2
K 11
T 20
C 3
L 12
U 21
D 4
M 13
V 22
E 5
N 14
W23
F 6
O 15
X 24
6
21
14
0
13
1
20
8
G 7
P 16
Y 25
F
U
N
_
M
A
T
H
H 8
Q 17
Z 26
How do we find the inverse???
• 1st you find what is called the determinant
• The determinant not only determines whether the
inverse of a matrix exists, but also influences
what elements the inverse contains
• For the matrix shown below, the determinant is
equal to ad – bc
• In other words, multiply the elements in each
diagonal, then subtract the products!
a b 
c d 


More about determinants
• If the determinant of a matrix equals zero, then
the inverse of that matrix does not exist!
• Every square matrix has a determinant, however
2 x 2 matrices are the only ones we will calculate
determinants for by hand
• For larger matrices, finding the determinant is
considerably more complicated (if you take a
linear programming course in college, or AP
Physics here at WHS, you may learn how to find
3 x 3 determinants by hand)
Finding the inverse of a 2 x 2 matrix
• For the matrix:
• The inverse is found
by calculating:
a b 
c d 


1  d b 


ad  bc  c a 
In other words:
-Switch the elements a and d
-Reverse the signs of the elements b and c
-Multiply by the fraction (1 / determinant)
Find the inverse of the matrix, if it exists.
Find the value of the determinant.
Since the determinant is not equal to 0, S –1 exists.
Definition of
inverse
a = –1, b = 0,
c = 8, d = –2
Answer:
Simplify.
Check:
Find the inverse of the matrix, if it exists.
Find the value of the determinant.
Answer: Since the determinant equals 0, T –1 does
not exist.
Find the inverse of each matrix, if it exists.
a.
Answer: No inverse exists.
b.
Answer:
Finding inverses for larger matrices
• We will not calculate inverses of 3 x 3 or larger
matrices by hand
• However, we CAN find these using the TI-83
• Enter your matrix using the EDIT menu, then
print it on your TI screen using the NAMES
menu
• Now hit the “X-1” button to indicate that you want
to find the inverse of this matrix!
• Let’s try some examples on the TI-83!!