5-5 Inverse of a Square Matrix
(D) Rank the players from strongest to weakest. Explain
the reasoning behind your ranking.
68. Dominance Relation. Each member of a chess team plays
one match with every other player. The results are given in
the table.
Player
Defeated
1. Anne
Diane
2. Bridget
Anne, Carol, Diane
3. Carol
Anne
4. Diane
Carol, Erlene
5. Erlene
Anne, Bridget, Carol
391
(A) Express the outcomes as an incidence matrix A by
placing a 1 in the ith row and jth column of A if
player i defeated player j and a 0 otherwise (see
Problem 63).
(B) Compute the matrix B ⫽ A ⫹ A2.
(C) Discuss matrix multiplication methods that can be
used to find the sum of the rows in B. State the
matrices that can be used and perform the necessary
operations.
(D) Rank the players from strongest to weakest. Explain
the reasoning behind your ranking.
Section 5-5 Inverse of a Square Matrix
Identity Matrix for Multiplication
Inverse of a Square Matrix
Application: Cryptography
In this section we introduce the identity matrix and the inverse of a square matrix.
These matrix forms, along with matrix multiplication, are then used to solve some
systems of equations written in matrix form in Section 5-6.
Identity Matrix for Multiplication
We know that for any real number a
(1)a ⫽ a(1) ⫽ a
The number 1 is called the identity for real number multiplication. Does the set
of all matrices of a given dimension have an identity element for multiplication?
That is, if M is an arbitrary m ⫻ n matrix, does M have an identity element I
such that IM ⫽ MI ⫽ M? The answer in general is no. However, the set of all
square matrices of order n (matrices with n rows and n columns) does have an
identity.
DEFINITION
1
IDENTITY MATRIX
The identity matrix for multiplication for the set of all square matrices
of order n is the square matrix of order n, denoted by I, with 1s along
the principal diagonal (from upper left corner to lower right corner) and
0s elsewhere.
392
5 SYSTEMS; MATRICES
For example,
FIGURE 1
Identity matrices.
冤
1
0
冥
0
1
冤
1
0
0
and
0
1
0
0
0
1
冥
are the identity matrices for all square matrices of order 2 and 3, respectively.
Most graphing utilities have a built-in command for generating the identity
matrix of a given order (see Fig. 1).
EXAMPLE
Identity Matrix Multiplication
1
1
冥冤
冥冤
(C)
冤
(D)
冤ad
b c
e f
(B)
MATCHED PROBLEM
冤
冤
0
1
0
b
e
h
0
1
(A)
1
0
0
a
d
g
1
0
0
0
1
c
f
i
a
d
冥冤
冥
a
d
g
1
0
0
b
e
1
0
0
冤
b
e
h
0
1
0
c
f
0
1
0
Multiply:
1 0 3 ⫺5
(A)
0 1 4
6
1 0 0 5 ⫺7
(B) 0 1 0 2
4
0 0 1 6 ⫺8
冤
冤
冥冤
冥
冥冤 冥
冥 冤
冥 冤
c
a
f ⫽ d
i
g
0
a
0 ⫽ d
1
g
a b
⫽
d e
0
a
0 ⫽
d
1
冥 冤
冥
and
and
冤
冤34
b
e
h
b
e
h
c
f
c
f
i
c
f
i
冥
冥
冥
冥
b c
e f
⫺5 1 0
6 0 1
5 ⫺7
1 0
2
4
0 1
6 ⫺8
冥冤
冤 冥冤
冥
冥
In general, we can show that if M is a square matrix of order n and I is the
identity matrix of order n, then
IM ⴝ MI ⴝ M
If M is an m ⫻ n matrix that is not square (m ⫽ n), then it is still possible
to multiply M on the left and on the right by an identity matrix, but not with the
same-size identity matrix (see Example 1, parts C and D). To avoid the complications involved with associating two different identity matrices with each nonsquare matrix, we restrict our attention in this section to square matrices.
5-5 Inverse of a Square Matrix
Explore/Discuss
1
393
The only real number solutions to the equation x2 ⫽ 1 are x ⫽ 1 and
x ⫽ ⫺1.
0 1
(A) Show that A ⫽
satisfies A2 ⫽ I, where I is the 2 ⫻ 2
1 0
identity.
0 ⫺1
(B) Show that B ⫽
satisfies B2 ⫽ I.
⫺1
0
(C) Find a 2 ⫻ 2 matrix with all elements nonzero whose square is the
2 ⫻ 2 identity matrix.
冤
冤
冥
冥
Inverse of a Square Matrix
In the set of real numbers, we know that for each real number a, except 0, there
exists a real number a⫺1 such that
a⫺1a ⫽ 1
The number a⫺1 is called the inverse of the number a relative to multiplication,
or the multiplicative inverse of a. For example, 2⫺1 is the multiplicative inverse
of 2, since 2⫺1(2) ⫽ 1. We use this idea to define the inverse of a square matrix.
DEFINITION
2
INVERSE OF A SQUARE MATRIX
If M is a square matrix of order n and if there exists a matrix M⫺1 (read
“M inverse”) such that
M⫺1M ⫽ MM⫺1 ⫽ I
then M⫺1 is called the multiplicative inverse of M or, more simply, the
inverse of M.
The multiplicative inverse of a nonzero real number a also can be written as
1/a. This notation is not used for matrix inverses.
Let’s use Definition 2 to find M⫺1, if it exists, for
M⫽
冤21 32冥
We are looking for
M⫺1 ⫽
冤ab cd冥
such that
MM⫺1 ⫽ M⫺1M ⫽ I
394
5 SYSTEMS; MATRICES
Thus, we write
M⫺1
M
I
2 3 a c
1 0
冤1 2冥 冤b d冥 ⫽ 冤0 1冥
and try to find a, b, c, and d so that the product of M and M⫺1 is the identity
matrix I. Multiplying M and M⫺1 on the left side, we obtain
⫹ 3b)
冤(2a
(a ⫹ 2b)
(2c ⫹ 3d)
1
⫽
(c ⫹ 2d)
0
冥 冤
冥
0
1
which is true only if
2a ⫹ 3b ⫽ 1
2c ⫹ 3d ⫽ 0
a ⫹ 2b ⫽ 0
c ⫹ 2d ⫽ 1
Solving these two systems, we find that a ⫽ 2, b ⫽ ⫺1, c ⫽ ⫺3, and d ⫽ 2.
Thus,
M⫺1 ⫽
冤⫺12
⫺3
2
冥
as is easily checked:
M⫺1
M
冤
2
1
冥冤
3
2
2
⫺1
M⫺1
I
⫺3
1
⫽
2
0
冥 冤
冥 冤
0
2
⫽
1
⫺1
M
⫺3
2
冥冤
2
1
冥
3
2
Unlike nonzero real numbers, inverses do not always exist for nonzero square
matrices. For example, if
N⫽
冤24 12冥
then, proceeding as before, we are led to the systems
2a ⫹ b ⫽ 1
2c ⫹ d ⫽ 0
4a ⫹ 2b ⫽ 0
4c ⫹ 2d ⫽ 1
These systems are both inconsistent and have no solution. Hence, N⫺1 does not
exist.
Being able to find inverses, when they exist, leads to direct and simple solutions to many practical problems. In the next section, for example, we will show
how inverses can be used to solve systems of linear equations.
The method outlined above for finding the inverse, if it exists, gets very
involved for matrices of order larger than 2. Now that we know what we are looking for, we can use augmented matrices, as in Section 5-3, to make the process
more efficient. Details are illustrated in Example 2.
5-5 Inverse of a Square Matrix
EXAMPLE
2
Finding an Inverse
Find the inverse, if it exists, of
冤
1
M⫽ 0
2
Solution
395
⫺1
2
3
1
⫺1
0
冥
We start as before and write
M⫺1
M
冤
1
0
2
⫺1
2
3
1
⫺1
0
I
冥冤
冥 冤
a d g
1
b e h ⫽ 0
c f i
0
0
1
0
0
0
1
冥
This is true only if
a⫺ b⫹c⫽1
d⫺ e⫹f⫽0
g⫺ h⫹i⫽0
2b ⫺ c ⫽ 0
2e ⫺ f ⫽ 1
2h ⫺ i ⫽ 0
2a ⫹ 3b
⫽0
2d ⫹ 3e
⫽0
2g ⫹ 3h
⫽1
Now we write augmented matrices for each of the three systems:
First
冤
1
0
2
⫺1
2
3
1
⫺1
0
ⱍ冥
Second
⫺1
2
3
冤
1
0
0
1
0
2
1
⫺1
0
ⱍ冥
0
1
0
Third
冤
1
0
2
⫺1
2
3
1
⫺1
0
ⱍ冥
0
0
1
Since each matrix to the left of the vertical bar is the same, exactly the same row
operations can be used on each augmented matrix to transform it into a reduced
form. We can speed up the process substantially by combining all three augmented
matrices into the single augmented matrix form
冤
1
0
2
⫺1
2
3
1
⫺1
0
ⱍ
1
0
0
0
1
0
冥
0
0 ⫽ 关M I兴
1
ⱍ
(1)
We now try to perform row operations on matrix (1) until we obtain a row-equivalent matrix that looks like matrix (2):
I
冤
1
0
0
0
1
0
ⱍ
B
冥
0 a d g
0 b e h ⫽ 关I B兴
1 c f i
ⱍ
(2)
If this can be done, then the new matrix to the right of the vertical bar is M⫺1!
Now let’s try to transform matrix (1) into a form like that of matrix (2). We follow
396
5 SYSTEMS; MATRICES
the same sequence of steps as in the solution of linear systems by Gauss–Jordan
elimination (see Section 5-3):
M
ⱍ
ⱍ
ⱍ
I
⫺1
1 1
2 ⫺1 0
3
0 0
冤
冤
冤
冤
冤
冤
1
0
2
0
1
0
0
0
1
冥
(⫺2)R1 ⫹ R3 → R3
1
⬃ 0
0
⫺1
1
1
2 ⫺1
0
5 ⫺2 ⫺2
1
⬃ 0
0
⫺1
1
1 0 0
1 ⫺ 12
0 12 0
5 ⫺2 ⫺2 0 1
1
⬃ 0
0
0
1
0
1
⬃ 0
0
0
1
0
1
⬃ 0
0
0
1
0
1
2
⫺ 12
1
2
1
2
⫺ 12
ⱍ
ⱍ
0
1
0
1
2
1
2
⫺ 52
1
2
1
2
1
0
⫺2
1
0
1 ⫺4 ⫺5
ⱍ
0
3
0 ⫺2
1 ⫺4
3
⫺2
⫺5
冥
冥
冥
冥
冥
0
0
1
0
0
1
0
0
2
1
2 R2
→ R2
R2 ⫹ R1 → R1
(⫺5)R2 ⫹ R3 → R3
2R3 → R3
(⫺ 12)R3 ⫹ R1 → R1
1
2 R3 ⫹ R2 → R2
⫺1
1 ⫽ 关I B兴
2
ⱍ
Converting back to systems of equations equivalent to our three original systems
(we won’t have to do this step in practice), we have
a ⫽ ⫺3
d ⫽ ⫺3
g ⫽ ⫺1
b ⫽ ⫺2
e ⫽ ⫺2
h ⫽ ⫺1
c ⫽ ⫺4
f ⫽ ⫺5
i ⫽ ⫺2
And these are just the elements of M⫺1 that we are looking for! Hence,
冤
3
M⫺1 ⫽ ⫺2
⫺4
3
⫺2
⫺5
⫺1
1
2
冥
Note that this is the matrix to the right of the vertical line in the last augmented
matrix.
Check
Since the definition of matrix inverse requires that
M⫺1M ⫽ I
and
MM⫺1 ⫽ I
(3)
it appears that we must compute both M⫺1M and MM⫺1 to check our work. However, it can be shown that if one of the equations in (3) is satisfied, then the other
5-5 Inverse of a Square Matrix
397
is also satisfied. Thus, for checking purposes it is sufficient to compute either
M⫺1M or MM⫺1—we don’t need to do both.
冤
3
M M ⫽ ⫺2
⫺4
⫺1
MATCHED PROBLEM
2
⫺1
1
2
3
⫺2
⫺5
冥冤
1
0
2
⫺1
2
3
冥 冤
1
1
⫺1 ⫽ 0
0
0
0
1
0
冥
0
0 ⫽I
1
3 ⫺1 1
Let M ⫽ ⫺1
1 0
1
0 1
(A) Form the augmented matrix 关M I兴.
(B) Use row operations to transform 关M I兴 into 关I B兴.
(C) Verify by multiplication that B ⫽ M⫺1.
冤
冥
ⱍ
ⱍ
ⱍ
The procedure used in Example 2 can be used to find the inverse of any square
matrix, if the inverse exists, and will also indicate when the inverse does not exist.
These ideas are summarized in Theorem 1.
THEOREM
1
Explore/Discuss
2
EXAMPLE
3
INVERSE OF A SQUARE MATRIX M
If 关M ⱍ I兴 is transformed by row operations into 关I ⱍ B兴, then the resulting
matrix B is M⫺1. If, however, we obtain all 0s in one or more rows to
the left of the vertical line, then M⫺1 does not exist.
(A) Suppose that the square matrix M has a row of all zeros. Explain
why M has no inverse.
(B) Suppose that the square matrix M has a column of all zeros. Explain
why M has no inverse.
Finding a Matrix Inverse
Find M⫺1, given M ⫽
⬃
⬃
ⱍ
ⱍ
4
⫺6
⫺1 1
2 0
0
1
冤⫺61
⫺ 14 14
2 0
0
1
冤
Solution
冤10
⫺ 14
1
2
ⱍ
1
4
3
2
冥
冥
冥
0
1
冤⫺64
1
4 R1
⫺1
2
冥
→ R1
6R1 ⫹ R2 → R2
2R2 → R2
398
5 SYSTEMS; MATRICES
⬃
⬃
ⱍ
冤
1
0
⫺ 14 14
1 3
0
2
冤
1
0
0 1 12
1 3 2
冥
ⱍ
1
4 R2
冥
⫹ R1 → R1
Thus,
M⫺1 ⫽
MATCHED PROBLEM
冤
1
2
1
3
2
冥
Find M⫺1, given M ⫽
3
EXAMPLE
4
Solution
Check by showing M⫺1M ⫽ I.
⫺6
⫺2
冤21
冥
Finding an Inverse
Find M⫺1, if it exists, given M ⫽
冤⫺510
ⱍ
⫺2 1
1 0
冥 冤
0
1
⬃
1
⫺5
⬃
冤
1
0
ⱍ
ⱍ
⫺2
1
冤⫺510
冥
1
⫺ 15 10
1 0
0
1
⫺ 15
0
0
1
1
10
1
2
冥
冥
We have all 0s in the second row to the left of the vertical line. Therefore, M⫺1
does not exist.
MATCHED PROBLEM
Find M⫺1, if it exists, given M ⫽
4
冤⫺26
⫺3
1
冥
Most graphing utilities can compute matrix inverses and can identify those
matrices that do not have inverses. A matrix that does not have an inverse is often
referred to as a singular matrix. Figure 2 illustrates the procedure on a graphing
utility. Note that the inverse operation is performed by pressing the x⫺1 key. Entering [A]^(⫺1) results in an error message.
FIGURE 2
Finding matrix inverses on a
graphing utility.
(a) Example 3
(b) Example 4
399
5-5 Inverse of a Square Matrix
Application: Cryptography
Matrix inverses can be used to provide a simple and effective procedure for
encoding and decoding messages. To begin, we assign the numbers 1 to 26 to the
letters in the alphabet, as shown below. We also assign the number 27 to a blank
to provide for space between words. (A more sophisticated code could include
both uppercase and lowercase letters and punctuation symbols.)
A
B
C
D
E
F
G
H
I
J
K
L
M
N
1
2
3
4
5
6
7
8
9
10
11
12
13
14
O
P
Q
R
S
T
U
V
W
X
Y
Z Blank
15
16
17
18
19
20
21
22
23
24
25
26
27
Thus, the message I LOVE MATH corresponds to the sequence
9
27
12
15
22
5
27
13
1
20
8
Any matrix whose elements are positive integers and whose inverse exists can be
used as an encoding matrix. For example, to use the 2 ⫻ 2 matrix
A⫽
冤45 34冥
to encode the above message, first we divide the numbers in the sequence into
groups of 2 and use these groups as the columns of a matrix B with two rows:
B⫽
冤279
12
15
22
5
27
13
1
20
冥
8
27
Proceed down the columns, not
across the rows.
(Notice that we added an extra blank at the end of the message to make the
columns come out even.) Then we multiply this matrix on the left by A:
AB ⫽
⫽
冤5 4冥 冤27
9
12
15
22
5
冤117
153
93
120
103
130
147
187
4
3
27
13
64
85
1
20
冥
8
27
113
148
冥
The coded message is
117 153
93
120 103 130 147 187
64
85
113 148
This message can be decoded simply by putting it back into matrix form and multiplying on the left by the decoding matrix A⫺1. Since A⫺1 is easily determined
if A is known, the encoding matrix A is the only key needed to decode messages
encoded in this manner. Although simple in concept, codes of this type can be
very difficult to crack.
400
5 SYSTEMS; MATRICES
EXAMPLE
Cryptography
5
The message
31 54 69 37 64 82 7 34 58 51 69 75 23 30 36 65 84 84
was encoded with the matrix A shown below. Decode this message.
冤
0
A⫽ 1
2
2
2
1
1
1
1
冥
We begin by entering the 3 ⫻ 3 encoding matrix A (Fig. 3). Then we enter the
coded message in the columns of a matrix C with three rows (Fig. 3). If B is the
matrix containing the uncoded message, then B and C are related by C ⫽ AB. To
find B, we multiply both sides of the equation C ⫽ AB by A⫺1 (Fig. 4).
Solution
FIGURE 3
FIGURE 4
Writing the numbers in the columns of this matrix in sequence and using the
correspondence between numbers and letters noted earlier produces the decoded
message:
23
8
15
W
H
O
27
9
19
I
S
27
3
1
18
12
C
A
R
L
27
7
1
21
19
19
G
A
U
S
S
27
The answer to this question can be found earlier in this chapter.
MATCHED PROBLEM
5
The message
46 84 85 55 101 100 59 95 132 25 42 53 52 91 90 43 71 83 19 37 25
was encoded with the matrix A shown below. Decode this message.
冤
1
A⫽ 2
2
1
1
3
1
2
1
冥
401
5-5 Inverse of a Square Matrix
Answers to Matched Problems
1. (A)
冤
5 ⫺7
(B) 2
4
6 ⫺8
1 1 0 0
0 0 1 0
1 0 0 1
⫺5
6
冥
3
4
冤 冥
冥 冤
ⱍ
3 ⫺1
2. (A) ⫺1
1
1
0
⫺1 3
3.
4. Does not exist
⫺ 12 1
冤
冤
冥
1
(B) 0
0
0
1
0
ⱍ
1
0
1
0
1 ⫺1
1
2
⫺1
⫺1
⫺1
2
冥
15.
A
3.
冤24
⫺3
5
1
5. 0
0
冤
冤
冤
冤
0
1
0
0
0
1
1
6. 0
0
0
1
0
0
0
1
7.
⫺2
2
5
⫺3
5
冥
2.
1 0 4
冤0 1冥 冤0
冥 冤10 01冥
4.
冤40
⫺2
2
5
1
4
1
⫺3
1
2
0
1
⫺1
2
5
7
1
0
0
0
1
0
0
0
1
1
0
0
0
1
0
0
0
1
冥冤
冥冤
1
3
4 ⫺2
1
0
⫺3
0
1
1
8.
2 ⫺1
2
5
7
3
⫺2
0
冥冤
冥冤
⫺3
2
冥
⫺3
2
冥 冤10 01冥
冥
冥
冥
冥
9.
冤
⫺4
3
;
3
2
冥冤
11.
冤⫺12
13.
2
3
;
冤⫺5
⫺8 3冥 冤 8
4
3
冥
冥冤
冥
2
1
;
⫺1
⫺1
1
⫺1
⫺2
⫺5
冥
⫺1
⫺1
2
⫺1
1
0
冥冤
3
⫺1
1
冥 冤
1
1
0 ⫽ 0
1
0
0
1
0
0
0
1
冥
冥冤
冥冤
冥冤
冥冤
2
1
⫺1
0
1
0
1
17. 0
2
⫺1
2
3
1
18. 3
0
0
1
0
⫺2
1
⫺1
0 1
0 ; 0
1 1
1 1
⫺2 ; 3
1 0
0
1
0
1
3
⫺1 ; ⫺2
0 ⫺4
⫺1
1
⫺1 ; ⫺3
0
0
0
0
0
⫺1
⫺1
1
⫺1
1
2
3
⫺2
⫺5
0
1
0
冥
冥
⫺1
⫺2
1
冥
冥
B
Given M in Problems 19–28, find M⫺1, and show that
M⫺1M ⫽ I.
In Problems 9–18, examine the product of the two matrices to
determine if each is the inverse of the other.
3
⫺2
冤
冤
冤
冤
1
0
⫺1
1
16. ⫺3
0
Perform the indicated operations in Problems 1–8.
冤10 01冥 冤24
1
2
⫺1
5. WHO IS WILHELM JORDAN
EXERCISE 5-5
1.
冤
1
1
⫺1
(C)
10.
冤
⫺2
⫺4
⫺1
1
;
2
2
⫺1
⫺2
12.
冤⫺2
⫺7
3
;
3
2
7
5
14.
冤⫺57
4
3
;
⫺3
⫺5
5
冥冤
冥冤
冥冤
冥
冥
⫺1
4
19.
冤01
冥
22.
冤25 13冥
1
25. 0
2
⫺2
1
⫺1
1
27. 0
1
1
2
0
冤
冤
0
1
2
0
⫺1
1
冥
冥
20.
冤⫺10
23.
冤12 37冥
冥
5
⫺1
冤
冤
21.
1 2
冤1 3冥
24.
冤21 11冥
冥
1
26. 1
0
3
2
⫺1
0
3
2
1
28. 2
1
0
⫺1
1
⫺1
0
⫺4
冥
Find the inverse of each matrix in Problems 29–32, if it exists.
冥
4
⫺7
29.
冤32 96冥
30.
冤⫺32
⫺4
6
冥
402
31.
5 SYSTEMS; MATRICES
冤23 35冥
32.
冤⫺54
冥
4
⫺3
46. Based on your observations in Problem 45, which of the
following is a true statement? Give a mathematical argument to support your conclusion.
C
(A) (AB)⫺1 ⫽ A⫺1B⫺1
Find the inverse of each matrix in Problems 33–38, if it exists.
2
0
33.
⫺1
2
4
⫺2
⫺1
⫺1
1
2
1
⫺1
1
1
⫺1
1
0
0
35.
冤
冤
冤
1
37. 0
1
5
1
6
10
4
15
冥
冤
冤
冤
4
1
34.
⫺3
冥
冥
⫺1
⫺1
1
2
1
⫺1
1
36. 2
0
⫺1
⫺1
1
0
1
1
1
38. 0
1
⫺5
1
⫺4
⫺10
6
⫺3
冥
冥
冥
冤a0 0d冥
冤a0 bd冥
冤⫺43
冥
2
⫺3
(B) A ⫽
冤⫺23
⫺1
2
冥
42. Based on your observations in Problem 41, if A ⫽ A⫺1 for
a square matrix A, what is A2? Give a mathematical argument to support your conclusion.
43. Find (A⫺1)⫺1 for each of the following matrices.
(A) A ⫽
冤
4
1
冥
2
3
(B) A ⫽
冤
冥
5
⫺1
5
3
44. Based on your observations in Problem 43, if A⫺1 exists
for a square matrix A, what is (A⫺1)⫺1? Give a mathematical argument to support your conclusion.
45. Find (AB)⫺1, A⫺1B⫺1, and B⫺1A⫺1 for each of the following pairs of matrices.
冤 冥
1 ⫺1
(B) A ⫽ 冤
2
3冥
3
(A) A ⫽
2
4
3
冤31 52冥
48. Cryptography. Encode the message FOX IN SOCKS
with the matrix A given above.
49. Cryptography. The following message was encoded with
the matrix A given above. Decode this message.
111 43 40 15 177 68
29 62 22 121 43 68
50 19 116 45
27
86
99 38 154 58 115 43 121 43 20
56 86 29 196 73 99 38
7
149
Problems 51–54 refer to the encoding matrix
41. Find A⫺1 and A2 for each of the following matrices.
(A) A ⫽
Problems 47–50 refer to the encoding matrix A ⫽
50. Cryptography. The following message was encoded with
the matrix A given above. Decode this message.
40. Discuss the existence of M⫺1 for 2 ⫻ 2 upper triangular
matrices of the form
M⫽
APPLICATIONS
47. Cryptography. Encode the message CAT IN THE HAT
with the matrix A given above.
39. Discuss the existence of M⫺1 for 2 ⫻ 2 diagonal matrices
of the form
M⫽
(B) (AB)⫺1 ⫽ B⫺1A⫺1
and
and
冤
3
B⫽
2
冤
6
B⫽
2
冥
7
5
冥
2
1
冤
1
0
B⫽ 2
0
1
0
1
1
0
1
1
1
1
1
1
0
0
1
0
2
冥
1
3
1
2
1
51. Cryptography. Encode the message DWIGHT DAVID
EISENHOWER with the matrix B given above.
52. Cryptography. Encode the message JOHN FITZGERALD KENNEDY with the matrix B given above.
53. Cryptography. The following message was encoded with
the matrix B given above. Decode this message.
41 84 82 44 74
54 89 39 102 44
136 81 149
25
67
56 67
86 44
20
90
54 43
68 135
54. Cryptography. The following message was encoded with
the matrix B given above. Decode this message.
22 15 57 5
80 87 53 96
136 81 149
47
51
54 58 89 45 84 46
68 116 39 113 68 135