Encode the message ​"JJ SYLVESTERJJ SYLVESTER​" using matrix A. Remember to partition the numerical message into groups of two.
Aequals=left bracket Start 2 By 2 Matrix 1st Row 1st Column 1 2nd Column 2 2nd Row 1st Column 1 2nd Column 3 EndMatrix right bracket 1	2	1	3
Question content area bottom
Part 1
The coded message​ is:   enter your response here  enter your response here  enter your response here  enter your response here  enter your response here  enter your response here  enter your response here  enter your response here  enter your response here  enter your response here  enter your response here  enter your response here

To encode the message "JJ SYLVESTERJJ SYLVESTER" using matrix \( A = \begin{bmatrix} 1 & 2 \\ 1 & 3 \end{bmatrix} \), we’ll proceed with the following steps:

1. **Convert each character to a numerical value**: We’ll use the A=1, B=2, ..., Z=26 encoding.
   
   For "JJ SYLVESTERJJ SYLVESTER":
   
   - **J = 10**
   - **S = 19**
   - **Y = 25**
   - **L = 12**
   - **V = 22**
   - **E = 5**
   - **S = 19**
   - **T = 20**
   - **E = 5**
   - **R = 18**

   Repeated for "JJ SYLVESTERJJ SYLVESTER."

2. **Partition the numerical message into pairs**: Convert "JJ SYLVESTERJJ SYLVESTER" as follows:

   - For the first "JJ SYLVESTER": \[(10, 10), (19, 25), (12, 22), (5, 19), (20, 5), (18, 10)\]
   - Repeat the same for the second "JJ SYLVESTER."

3. **Multiply each pair by Matrix \( A \)**:

   Each vector \(\begin{bmatrix} x \\ y \end{bmatrix}\) will be multiplied by matrix \( A \):

   \[
   A \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 1 & 2 \\ 1 & 3 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} x + 2y \\ x + 3y \end{bmatrix}
   \]

   Let’s calculate each encoding.

---

### Calculations for Encoding

**1. Encoding (10, 10):**

\[
\begin{bmatrix} 1 & 2 \\ 1 & 3 \end{bmatrix} \begin{bmatrix} 10 \\ 10 \end{bmatrix} = \begin{bmatrix} 10 + 20 \\ 10 + 30 \end{bmatrix} = \begin{bmatrix} 30 \\ 40 \end{bmatrix}
\]

**2. Encoding (19, 25):**

\[
\begin{bmatrix} 1 & 2 \\ 1 & 3 \end{bmatrix} \begin{bmatrix} 19 \\ 25 \end{bmatrix} = \begin{bmatrix} 19 + 50 \\ 19 + 75 \end{bmatrix} = \begin{bmatrix} 69 \\ 94 \end{bmatrix}
\]

**3. Encoding (12, 22):**

\[
\begin{bmatrix} 1 & 2 \\ 1 & 3 \end{bmatrix} \begin{bmatrix} 12 \\ 22 \end{bmatrix} = \begin{bmatrix} 12 + 44 \\ 12 + 66 \end{bmatrix} = \begin{bmatrix} 56 \\ 78 \end{bmatrix}
\]

**4. Encoding (5, 19):**

\[
\begin{bmatrix} 1 & 2 \\ 1 & 3 \end{bmatrix} \begin{bmatrix} 5 \\ 19 \end{bmatrix} = \begin{bmatrix} 5 + 38 \\ 5 + 57 \end{bmatrix} = \begin{bmatrix} 43 \\ 62 \end{bmatrix}
\]

**5. Encoding (20, 5):**

\[
\begin{bmatrix} 1 & 2 \\ 1 & 3 \end{bmatrix} \begin{bmatrix} 20 \\ 5 \end{bmatrix} = \begin{bmatrix} 20 + 10 \\ 20 + 15 \end{bmatrix} = \begin{bmatrix} 30 \\ 35 \end{bmatrix}
\]

**6. Encoding (18, 10):**

\[
\begin{bmatrix} 1 & 2 \\ 1 & 3 \end{bmatrix} \begin{bmatrix} 18 \\ 10 \end{bmatrix} = \begin{bmatrix} 18 + 20 \\ 18 + 30 \end{bmatrix} = \begin{bmatrix} 38 \\ 48 \end{bmatrix}
\]

Repeat these results for the second "JJ SYLVESTER".

---

### Final Encoded Message

The coded message is:

\[
30, 40, 69, 94, 56, 78, 43, 62, 30, 35, 38, 48, 30, 40, 69, 94, 56, 78, 43, 62, 30, 35, 38, 48
\]

Would you like to see step-by-step details on encoding each pair or have any questions?

---

**Relative Questions:**

1. How does matrix multiplication work in cryptography?
2. Why is partitioning the message into pairs essential in this encoding method?
3. What would change in the encoding if a different matrix were used?
4. Can this encoding method be used to decode the message?
5. How is modular arithmetic relevant in matrix encoding for cryptography?

**Tip:** Always double-check each step in matrix encoding, as errors can propagate through the entire message.

Encode the message 'JJ SYLVESTERJJ SYLVESTER' using matrix A. Remember to partition the numerical message into groups of two.
A = [ [1, 2], [1, 3] ]
Part 1
The coded message​ is:    

Learn to encode the message 'JJ SYLVESTERJJ SYLVESTER' using a 2x2 matrix and matrix multiplication. This solution provides a step-by-step guide for character-to-number encoding, partitioning, and matrix transformations. Suitable for high school students exploring cryptography and matrix operations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation