Extra Credit If you choose to do any of this extra credit it is due on the day of the Final Exam. It will not be accepted late under any circumstances. 1. (8 points) Cryptography Using Matrices. First an example of a simple encryption technique using matrices. Alice wants to send a secret message to Bob using only capital letters. She assigns numbers to each of the letters in the alphabet, A = 0, B = 1, C = 2, ..., Z = 25. She then puts her secret message into a matrix A = ( 12 0 19 7 ) . Here the secret message is read, 12 0 19 7, in other words, MATH. Alice then multiplies A on the left by an invertible matrix B and sends BA to Bob, who has B−1. Once Bob receives Alice’s matrix he computes B−1(BA) = (B−1B)A = IA = A and so he has Alice’s matrix, and hence her secret message. In this problem, A is the plaintext, this is the unencrypted message. This is Alice’s message in the exam- ple. BA is the ciphertext, this is the encrypted message. In our example Alice sends BA to Bob. B−1 is the key, which is used to decode the encrypted message. Bob has the key in our example. So to recap, 1. Alice puts her message into a matrix A. 2. Alice multiplies A by an invertible matrix B and sends BA to Bob who has B−1. 3. Bob who has both B−1 and BA now can compute A since B−1(BA) = A. (a) Suppose Alice puts her secret message into a matrix A, multiplies it by B and sends BA = ( 57 114 91 178 ) to Bob who has B−1 = ( −3 2 5 2 −3 2 ) . What is Alice’s secret message? (b) As above suppose Alice puts her secret message into a matrix A, multiplies it by B and sends BA to Bob who has B−1. Unknown to Alice, Eve is aware of Alice’s encryption method and has intercepted the ciphertext BA = ( 1 2 3 1 ) . Moreover, Eve has managed to get B from Bob but she can only make out some of the numbers. She has B−1 = ( 1 2 3 x ) where she has determined that x is either 1, 2, or 6. List all the possible plaintexts. Hint: Be careful! 2. (4 points) Use mathematical induction to prove A = ( 2 0 0 3 )n = ( 2n 0 0 3n ) for every positive integer n. 3. (7 points) Use mathematical induction to prove (PDP −1)n = PDnP −1 for every positive integer n. Here P and D are matrices and P is invertible. ...

1. Extra Credit
If you choose to do any of this extra credit it is due on the day
of the Final Exam. It will not
be accepted late under any circumstances.
1. (8 points) Cryptography Using Matrices. First an example of
a simple encryption
technique using matrices. Alice wants to send a secret message
to Bob using only capital letters.
She assigns numbers to each of the letters in the alphabet, A =
0, B = 1, C = 2, ..., Z = 25.
She then puts her secret message into a matrix A =
(
12 0
19 7
)
. Here the secret message is
read, 12 0 19 7, in other words, MATH. Alice then multiplies A
on the left by an invertible
matrix B and sends BA to Bob, who has B−1. Once Bob
receives Alice’s matrix he computes
B−1(BA) = (B−1B)A = IA = A and so he has Alice’s matrix,
and hence her secret message.
In this problem,
A is the plaintext, this is the unencrypted message. This is
Alice’s message in the exam-

2. ple.
BA is the ciphertext, this is the encrypted message. In our
example Alice sends BA to
Bob.
B−1 is the key, which is used to decode the encrypted message.
Bob has the key in our
example.
So to recap,
1. Alice puts her message into a matrix A.
2. Alice multiplies A by an invertible matrix B and sends BA to
Bob who has B−1.
3. Bob who has both B−1 and BA now can compute A since
B−1(BA) = A.
(a) Suppose Alice puts her secret message into a matrix A,
multiplies it by B and sends
BA =
(
57 114
91 178
)
to Bob who has B−1 =
(
−3 2
5
2

3. −3
2
)
. What is Alice’s secret message?
(b) As above suppose Alice puts her secret message into a
matrix A, multiplies it by B and
sends BA to Bob who has B−1. Unknown to Alice, Eve is aware
of Alice’s encryption method
and has intercepted the ciphertext BA =
(
1 2
3 1
)
. Moreover, Eve has managed to get B from
Bob but she can only make out some of the numbers. She has
B−1 =
(
1 2
3 x
)
where she has
determined that x is either 1, 2, or 6. List all the possible
plaintexts. Hint: Be careful!

4. 2. (4 points) Use mathematical induction to prove A =
(
2 0
0 3
)n
=
(
2n 0
0 3n
)
for every
positive integer n.
3. (7 points) Use mathematical induction to prove (PDP −1)n =
PDnP −1 for every positive
integer n. Here P and D are matrices and P is invertible.