1. 4.4 Identity and Inverse Matrices
What is a 2 by 2 identity matrix? A 3 by 3?
How do you create an inverse matrix?
What symbol is used for an inverse matrix?
How do you encode or decode a message with a matrix?

2. Using Inverse Matrices
The number 1 is a multiplicative identity for real numbers because 1 • a = a and a • 1 = a. For matrices, the n  n identity matrix is the matrix that has 1’s on the main diagonal and 0’s elsewhere.
2  2 Identity Matrix
3  3 Identity Matrix
1 0 0
0 1 0
0 0 1
1 0
0 1
I =
I =
If A is any n  n matrix and I is the n  n identity matrix, then IA = A and AI = A.
Two n  n matrices are inverses of each other if their product (in both orders) is the n  n identity matrix. For example, matrices A and B below are inverses of each other.
3 –1
–5 2 =
2 1
5 3 =
1 0
0 1 I
2 1
5 3 =
3 –1
–5 2 =
1 0
0 1 I AB BA
The symbol used for the inverse of A is A –1.

3. Using Inverse Matrices
The number 1 is a multiplicative identity for real numbers because 1 • a = a and a • 1 = a. For matrices, the n  n identity matrix is the matrix that has 1’s on the main diagonal and 0’s elsewhere.
THE INVERSE OF A 2  2 MATRIX
The inverse of the matrix A = is
a b
c d
A –1 = = provided a d – c b  0.
d –b
–c a 1
| A|
a d – c b

4. Finding the Inverse of a 2  2 Matrix
Find the inverse of A =
3 1
4 2
SOLUTION 1 –2 2 3 =
2 –1
–4 3 1
6 – 4 =
2 –1
–4 3 1 2 =
A –1
CHECK
You can check the inverse by showing that AA –1 = I = A –1A.
3 1
4 2 1 –2 2 3 1 –2 2 3
3 1
4 2 =
1 0
0 1
and
1 0
0 1 =

5. Solving a Matrix Equation
Solve the matrix equation AX = B for the 2  2 matrix X. A B
4 –1
–3 1
8 –5
–6 3
X =
SOLUTION
Begin by finding the inverse of A.
1 1
3 4 1
4 – 3 = =
1 1
3 4
A –1

6. Solving a Matrix Equation
Solve the matrix equation AX = B for the 2  2 matrix X. A B
4 –1
–3 1
8 –5
–6 3
X =
To solve the equation for X, multiply both sides of the equation by A –1 on the left.
1 1
3 4
4 –1
–3 1 X =
8 –5
–6 3
1 1
3 4
A –1AX = A –1B
1 0
0 1 X =
2 –2
0 –3
IX = A –1B X =
2 –2
0 –3
X = A –1B
CHECK
You can check the solution by multiplying A and X to see if you get B.

7. Using Inverse Matrices
Some matrices do not have an inverse. You can tell whether a matrix has an inverse by evaluating its determinant.
If det A = 0, then A does not have an inverse. If det A  0, then A has an inverse.
The inverse of a 3  3 matrix is difficult to compute by hand. A calculator that will compute inverse matrices is useful in this case.

8. Using Inverse Matrices in Real Life
A cryptogram is a message written according to a secret code. (The Greek word kruptos means hidden and the Greek word gramma means letter.) The following technique uses matrices to encode and decode messages.
First assign a number to each letter in the alphabet with 0 assigned to a
blank space.
_ = 0
E = 5
J = 10
O = 15
T = 20
Y = 25
A = 1
F = 6
K = 11
P = 16
U = 21
Z = 26
B = 2
G = 7
L = 12
Q = 17
V = 22
C = 3
H = 8
M = 13
R = 18
W = 23
D = 4
I = 9
N = 14
S = 19
X = 24
Then convert the message to numbers partitioned into 1  2 uncoded row matrices.
To encode a message, choose a 2  2 matrix A that has an inverse and multiply the uncoded row matrices by A on the right to obtain coded row matrices.

9. Use the list below to convert the message GET HELP to row matrices.
Converting a Message
Use the list below to convert the message GET HELP to row matrices.
_ = 0
E = 5
J = 10
O = 15
T = 20
Y = 25
A = 1
F = 6
K = 11
P = 16
U = 21
Z = 26
B = 2
G = 7
L = 12
Q = 17
V = 22
C = 3
H = 8
M = 13
R = 18
W = 23
D = 4
I = 9
N = 14
S = 19
X = 24
SOLUTION G E T _ H L P
[ ]
[20 0]
[ ]
[ ]

10. CRYPTOGRAPHY Use A = to encode the message GET HELP. 2 3 –1 –2
Encoding a Message
CRYPTOGRAPHY Use A = to encode the message GET HELP.
2 3
–1 –2
SOLUTION
The coded row matrices are obtained by multiplying each of the uncoded row matrices from the previous example by the matrix A on the right.
UNCODED
ROW MATRIX
ENCODING
MATRIX A
CODED ROW
MATRIX
2 3
–1 –2
[7 5]
= [9 11]
The coded message is 9, 11, 40, 60, 11, 14, 8, 4.
2 3
–1 –2
[20 0]
= [40 60]
2 3
–1 –2
[8 5]
= [11 14]
2 3
–1 –2
[12 16]
= [8 4]

11. Decoding a Message
DECODING USING MATRICES For an authorized decoder who knows the matrix A, decoding is simple. The receiver only needs to multiply the coded row matrices by A –1 on the right to retrieve the uncoded row matrices.
CRYPTOGRAPHY Use the inverse of A = to decode this message:
3 –1
–2 1
SOLUTION
1 1
2 3 1
3 – 2
A –1 = =
1 1
2 3
First find A –1:
To decode the message, partition it into groups of two numbers to form coded row matrices. Then multiply each coded row matrix by A –1 on the right to obtain the uncoded row matrices.

12. CRYPTOGRAPHY Use the inverse of A = to decode this message: 3 –1 –2 1
Decoding a Message
CRYPTOGRAPHY Use the inverse of A = to decode this message:
3 –1
–2 1
CODED ROW MATRIX
DECODING MATRIX A –1
UNCODED ROW MATRIX
1 1
2 3
[– 4 3]
= [2 5]
1 1
2 3
[–23 12]
= [1 13]
1 1
2 3
[–26 13]
= [0 13]
1 1
2 3
[15 –5]
= [5 0]

13. CRYPTOGRAPHY Use the inverse of A = to decode this message: 3 –1 –2 1
Decoding a Message
CRYPTOGRAPHY Use the inverse of A = to decode this message:
3 –1
–2 1
CODED ROW MATRIX
DECODING MATRIX A –1
UNCODED ROW MATRIX
1 1
2 3
[31 –5]
= [ ]
1 1
2 3
[– ]
= [0 19]
1 1
2 3
[–21 12]
= [3 15]
1 1
2 3
[ ]
= [ ]

14. CRYPTOGRAPHY Use the inverse of A = to decode this message: 3 –1 –2 1
Decoding a Message
CRYPTOGRAPHY Use the inverse of A = to decode this message:
3 –1
–2 1
CODED ROW MATRIX
DECODING MATRIX A –1
UNCODED ROW MATRIX
1 1
2 3
[75 –25]
= [25 0]
The uncoded row matrices are as follows.
[2 5][1 13][0 13][5 0][ ][0 19][3 15][20 20][25 0]

15. Decoding a Message
CRYPTOGRAPHY Use the inverse of A = to decode this message:
3 –1
–2 1
_ = 0
E = 5
J = 10
O = 15
T = 20
Y = 25
A = 1
F = 6
K = 11
P = 16
U = 21
Z = 26
B = 2
G = 7
L = 12
Q = 17
V = 22
C = 3
H = 8
M = 13
R = 18
W = 23
D = 4
I = 9
N = 14
S = 19
X = 24
You can read the message as follows:
[2 5][1 13][0 13][5 0][ ][0 19][3 15][20 20][25 0] B E A M _ M E _ U P _ S C O T T Y _

16. What is a 2 by 2 identity matrix? A 3 by 3?
How do you create an inverse matrix?
What symbol is used for an inverse matrix?
How do you encode or decode a message with a matrix?

17. Assignment 4.4
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