Problem 78 - College Algebra

Question

Use a coding matrix \(A\) of your choice. Use a graphing utility to find the multiplicative inverse of your coding matrix. Write a cryptogram for cach message. Check your result by decoding the cryptogram. Use your graphing utility to perform all necessary matrix multiplications. $$\begin{array}{llllllllllll}\mathrm{A} & \mathrm{R} & \mathrm{T} & \- & \mathrm{E} & \mathrm{N} & \mathrm{R} & \mathrm{I} & \mathrm{C} & \mathrm{H} & \mathrm{E} & \mathrm{S} \\\1 & 18 & 20 & 0 & 5 & 14 & 18 & 9 & 3 & 8 & 5 & 19\end{array}$$

Step-by-Step Solution

Answer

The multiplication of matrix A and the numbers representing the letters of the message will give the encoded message. To decode this message, multiply the encoded message by the inverse of matrix A. The result will give the decoded message.

Step 1: Choose a Coding Matrix

Choose a coding matrix A. For this example, we'll use A = \[\begin{{bmatrix}}1 & 2 \\ 2 & 1 \end{{bmatrix}}\]. Please take note that the coding matrix should be invertible.

Step 2: Convert Text to Numbers

Convert the given alphabets to numbers using the given key. For example - \( A = 1, B = 2, .. , Z = 26, space = 0 \). The given message 'ART ENRICHES' translates to \[\begin{{pmatrix}}1 & 18 & 20 & 0 & 5 & 14 & 18 & 9 & 3 & 8 & 5 & 19 \end{{pmatrix}}\].

Step 3: Encode the Message with the Coding Matrix

Multiply the coding matrix with each pair of numbers that represents a letter in the message. This will create a code for each pair which we then string together to create the cryptogram.

Step 4: Find the Multiplicative Inverse

To decrypt the message, find the multiplicative inverse of the coding matrix A. You can achieve this by using the formula for 2x2 matrices where the inverse is 1/(ad-bc) times \[\begin{{bmatrix}}d & -b \\ -c & a \end{{bmatrix}}\]. In this case the inverse would be 1/(1*1 - 2*2) times \[\begin{{bmatrix}}1 & -2 \\ -2 & 1 \end{{bmatrix}}\] which simplifies to -1 times \[\begin{{bmatrix}}1 & -2 \\ -2 & 1 \end{{bmatrix}}\] or \[\begin{{bmatrix}}-1 & 2 \\ 2 & -1 \end{{bmatrix}}\].

Step 5: Decode the Cryptogram

Next, use the inverse matrix to multiply the code. The result of this multiplication would be the original pairs of numbers that represented the letters in the message. Convert these numbers back to letters using the key.

Key Concepts

Coding Matrix

A coding matrix plays a pivotal role in the process of matrix encryption. In essence, it is a prearranged, square matrix used to encode a plaintext message into ciphertext. Choosing a coding matrix requires a bit of caution. For the encoding and decoding process to function correctly, the matrix must be invertible, meaning there has to exist another matrix, called its inverse, that, when multiplied with the original, results in the identity matrix. The identity matrix is a special matrix where all the elements of the principal diagonal are ones, and all other elements are zeros. It acts much like the number one in regular multiplication.

In our exemplified encryption scenario, we opted for a simple 2x2 coding matrix
\[\begin{bmatrix}1 & 2 \ 2 & 1 \end{bmatrix}\]. This matrix must then be applied to the plaintext message structured in a numerical form to create an encoded message.

Multiplicative Inverse of a Matrix

The multiplicative inverse of a matrix, often simply referred to as the inverse matrix, is a cornerstone concept for matrix decryption. The idea is equivalent to the reciprocal of a number, but in the context of matrices. For a matrix to have an inverse, it must be square (same number of rows as columns) and must have a non-zero determinant. The determinant for a 2x2 matrix \(A = \begin{bmatrix}a & b \ c & d\end{bmatrix}\) is calculated as \(ad - bc\). If this value is zero, the matrix cannot be inverted.

To compute the inverse of a 2x2 matrix, you use the formula \(A^{-1} = \frac{1}{ad - bc} \times \begin{bmatrix}d & -b \ -c & a\end{bmatrix}\). For our selected coding matrix, the determinant is \(-3\), and its inverse is \(\begin{bmatrix}-1 & 2 \ 2 & -1\end{bmatrix}\), which becomes critical when reversing the encryption process.

Matrix Multiplication in Cryptography

Matrix multiplication is integral to the encryption and decryption processes of matrix cryptography. It might appear daunting at first glance, but it's actually straightforward with a bit of practice. When you're encoding a message, you multiply the coding matrix by each pair of numbers representing the plaintext message to generate encoded pairs, strings of numbers that form your cryptogram. This process transforms the understandable message into an obscured format that's nearly impossible to decipher without the correct decryption key.

Decryption involves a similar multiplication process but in reverse, using the inverse of the coding matrix. By performing matrix multiplication of the inverse with the cryptogram, we revert the encoded pairs back into the original numerical format of the plaintext message. This underlying multiplication operation ensures the security of communication, given that only the right inverse matrix can successfully decode the cryptogram.

Converting Text to Numbers for Encryption

Before a message can be encrypted using matrices, it needs to be converted from text to a numerical format. Each character, including spaces, is assigned a specific number. For instance, in our example, we use \(A = 1, B = 2, \ldots, Z = 26, \text{space} = 0\). This conversion is essential because matrices operate on numerical values, not letters. After encoding the message with the coding matrix, we end up with a string of numbers that represent the encrypted form of the original message, known as the cryptogram.

Upon decrypting, you'll need to reverse this conversion to convert the numerical code back into human-readable text. So, understanding this process of converting text to numbers is crucial for anyone delving into the realm of matrix encryption and decryption.