Problem 62 - Algebra and Trigonometry

Question

It's January \(1,\) and you've written down your major goal for the year. You do not want those closest to you to see what you've written in case you do not accomplish your objective. Consequently, you decide to use a coding matrix to encode your goal. Explain how this can be accomplished.

Step-by-Step Solution

Answer

To keep your goal a secret from others around you, it can be encoded using a chosen coding matrix. The goal is first turned into numerical format, which is then encoded using the chosen matrix. Only with the knowledge of the original matrix, the goal can be decoded back to the text.

Step 1: Understanding Matrix Encoding

Matrix encoding is a process where each letter of the message is converted into a numerical representation, usually Unicode, which is then encoded using a chosen matrix. For instance, we can use a simple numerical substitution where 'A' is 1, 'B' is 2, up to 'Z' being 26.

Step 2: Create coding matrix

Select a 2x2 invertible matrix. The matrix elements must be integers. This is going to be our secret coding matrix. Let it be \[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \]

Step 3: Encoding the text

Take the numerical representation of the first two letters of your goal, put them in a 2x1 matrix. Multiply your 2x2 matrix by your 2x1 matrix to encode the first two letters. Repeat with the next two letters until the entire goal is encoded.

Step 4: Decoding the text

Remember, only you have the secret matrix to decode the text. To get the original goal back, take the numerical representations of the encoded letters, multiply it by the inverse of the original matrix, and convert back to text.