Inverse of a Square Matrix
Inverse of a Square Matrix
Section 4-4
Prof. Nathan Wodarz
Math 109 - Fall 2008
Contents
1 Identity Matrix for Multiplication
2
Identity Matrix for Multiplication
. . . . . . . . . . . . . . . . .
2
3
Inverse of a Square Matrix . . . . . . . . . . . . . . . . . . . . .
3
6
Cryptography . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1
1 Identity Matrix for Multiplication
1.1
Identity Matrix for Multiplication
Identity Matrix for Multiplication
• The number 1 is called the identity for multiplication of real numbers since
1 · a
= a
= a · 1
• Is there a matrix with the same property?
• That is, is there a matrix I such that I M
=
M
=
MI for all matrices M ?
• Yes, if M is square (no otherwise)
Identity Matrix for Multiplication
• The identity element for multiplication for an n × n square matrix is a square matrix of size n with 1’s on the principal diagonal, and 0’s elsewhere.
• We write I for the identity matrix ( I n if we want to emphasize the size)
Example.
–
"
1 0
#
0 1
–
1 0 0
0 1 0
0 0 1
–
"
2 3
# "
1 0
#
4 5 0 1
=
"
2 3
#
4 5
2
Identity Matrix for Multiplication
Problem 1.
Find the identity matrix of the appropriate size.
"
2 3
#
5 1
A.
" 1 0 #
0 1
B.
C.
"
0 1
#
1 0
"
1 1
#
1 1
D.
" 1
2
1
5
1
3
1
#
E.
None of the above
2 Inverse of a Square Matrix
2.1
Inverse of a Square Matrix
Inverse of a Square Matrix
• If a , 0, the number
1 a satisfies a ·
1 a
=
1
=
1 a
· a
• We also write a − 1 =
1 a
• We say that a − 1 is the multiplicative inverse of a
• Does a matrix M have a multiplicative inverse M
− 1
? (We never write 1
/
M )
• Say “ M inverse”
• Need M
− 1 so M M
− 1 =
I
=
M
− 1 M
• Can’t happen if M isn’t square
• Even if M is square, may not happen
• If M doesn’t have an inverse, M is singular
Example.
The inverse of
"
5 4
4 3
# is
"
− 3 4
4 − 5
#
3
Finding the Inverse of a Square Matrix
• Amounts to solving a system with n
2 equations and n
2 variables
• Use augmented matrices
• Combine multiple augmented matrices into one.
• Write [ M | I ] and use row operations to change to reduced form
• Ideally, get h
I M
− 1 i
• M is invertible if we can reduce the left side to I
• Otherwise M is singular
Finding the Inverse of a Square Matrix
Problem 2.
Determine whether A is the inverse of B.
A
=
"
5 3
3 2
# and B
=
"
2 − 3
#
− 3 5
A.
Yes
B.
No
Finding the Inverse of a Square Matrix
Problem 3.
Which of the following matrices has an inverse?
A.
"
− 2 3
#
4 1
B.
C.
"
3 − 2 1
#
4 0 7
" 0 4
0 − 2
#
D.
0 − 1
3 5
− 1 3
E.
None of the above
4
Finding the Inverse of a Square Matrix
Problem 4.
Find the inverse, if it exists, of the given matrix.
"
5 8
#
3 5
A.
B.
"
− 5 − 8
#
− 3 − 5
"
− 5 3
8 − 5
#
C.
"
5 3
#
8 5
D.
" 5 − 8 #
− 3 5
E.
Does not exist
F.
None of the above
Finding the Inverse of a Square Matrix
Problem 5.
Find the inverse, if it exists, of the given matrix.
"
− 6 − 6
#
− 5 − 5
A.
5
−
11
5
−
11
6
−
11
6
−
11
B.
5
−
11
5
11
6
−
11
6
11
C.
Does not exist
D.
5
−
11
5
11
6
−
11
6
11
E.
5
11
5
11
6
11
6
11
F.
None of the above
5
Finding the Inverse of a Square Matrix
Problem 6.
Find the inverse, if it exists, of the given matrix.
0 4 4
− 2 0 8
0 2 0
A.
B.
− 1 −
1
2
1
−
1
2
0
0
4
0
1
4
1
1 − − 2
2
1
0
0 −
2
1
2
− 2
1
2
0
C.
Does not exist
D.
0
−
1
2
0
0
1
1
4
2
1
4
1
8
0
E.
1
−
1
2
− 2
0
0
1
2
1
−
4
0
1
2
F.
None of the above
3 Applications
3.1
Cryptography
Encrypting Messages
• Any English sentence can be converted to a string of numbers
6
• Use code: blank 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 W 23
F 6 O 15 X 24
G 7 P 16 Y 25
H 8 Q 17 Z 26
• Take a message and divide it into a matrix B with 2 rows
• Take a 2 × 2 matrix A (the encoding matrix ) and left-multiply to get AB
• This gives the coded message
Encrypting Messages
Example.
"
2 1
#
3 1
• Encode “A SECRET CODE” using the encoding matrix A
=
• The message becomes 1 0 19 5 3 18 5 20 0 3 15 4 5
• Write as a matrix as B
=
"
1 19 3 5 0 15 5
#
0 5 18 20 3 4 0
• Write a blank at the end to make columns come out correctly
• We get AB
=
"
2 43 24 30 3 34 10
#
3 62 27 35 3 49 15
• Encoded message is 2 3 43 62 24 27 30 35 3 3 34 49 10 15
Decrypting Messages
• Find inverse A
− 1 (the decoding matrix ) of encoding matrix A
• Left multiply by A
− 1
• Remaining matrix is the decoded message
• Can also use larger matrices
7
Decrypting Messages
Example.
• Decode the message 2 3 43 62 24 27 30 35 3 3 34 49 10 15 using the encoding matrix A
=
"
2 1
#
3 1
• The decoding matrix is A − 1 =
"
− 1 1
3 − 2
#
• Multiply
"
− 1 1
3 − 2
# "
2 43 24 30 3 34 10
#
3 62 27 35 3 49 15
Cryptography
Problem 7.
A message has been encoded and the matrix which the receiver gets is shown below.
The message is 3 11 0 40 45 65 54 60 66 22 21 17 60 30 6 4 15 29 0 38. The encoding matrix A which was used to encode the message is A
=
"
2 1 and use it to decode the message, assuming that the numerical assignment used was A
=
1 , B
=
2
, . . . ,
Z
=
26 , blank
=
0 .
0 3 #
. Find the decoding matrix A
− 1
A.
DRINK ENOUGH COKE
B.
DRINK ENOUGH MILK
C.
EAT YOUR BROCCOLI
D.
EAT YOUR VEGETABLES
E.
None of the above
Summary
Summary
You should be able to:
• Identify the identity matrix
• Find the inverse of a square matrix
• Use matrix inverses in cryptographic applications
8
