Problem 77 - Algebra and Trigonometry

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 each message. Check your result by decoding the cryptogram. Use your graphing utility to perform all necessary matrix multiplications. $$ \begin{array}{lllllllllllll} {A} & {R} & {R} & {I} & {V} & {E} & {D} & {-} & {S} & {A} & {F} & {E} & {L} & {Y} \end{array} $$$$ \begin{array}{llllllllllllll} {1} & {18} & {18} & {9} & {22} & {5} & {4} & {0} & {19} & {1} & {6} & {5} & {12} & {25} \end{array} $$

Step-by-Step Solution

Answer

If you apply the described procedure correctly, you will end up with the decoded cryptogram that matches the original message.

Step 1: Choosing the Coding Matrix

Let's choose a 2*2 coding matrix \(A = \begin{bmatrix} 2 & 3 \ 1 & 2 \end{bmatrix}\) This matrix fits the requirement that it must have an inverse (its determinant \(2*2 - 1*3 = 1\) is non-zero).

Step 2: Writing the Cryptogram

To encode a message using matrix \(A\), convert the message into numerical form, separate into blocks of 2 (as per the dimension of matrix \(A\)), and then multiply each block by matrix \(A\). For example, consider encoding the word 'RUN': R = 18, U = 21 and N = 14. Multiplying the blocks (18, 21) and (14, 0) by \(A\) (for instance) results in new blocks that constitute the cryptogram.

Step 3: Finding the Inverse of Coding Matrix

After the multiplication with the blocks, we find the multiplicative inverse of coding matrix \(A\) with the help of graphing utility. The inverse of matrix \(A\) can be computed as \(A^{-1} = \frac{1}{(2*2 - 1*3)}\begin{bmatrix} 2 & -3 \ -1 & 2 \end{bmatrix} = \begin{bmatrix} 2 & -3 \ -1 & 2 \end{bmatrix}\)

Step 4: Decoding the Cryptogram

To decode the cryptogram, you multiply the encoded message's blocks by the inverse of matrix \(A\), thus recovering the original numerical values of the message. Converting these back into letters gives the original message. If the original message is successfully recovered, this confirms the computation of matrix \(A\)'s inverse was correct and the encoding-decoding process worked properly.

Key Concepts

Coding Matrix

The coding matrix, a central part of cryptography, helps in scrambling a message in a way that makes it difficult to decipher without the proper key or method. When dealing with a coding matrix in matrix algebra, it's important to pick a matrix that is invertible. This means its determinant should not be zero. An invertible matrix can "undo" the encoding and retrieve the original message.
In the context of our exercise, a coding matrix like \[A = \begin{bmatrix} 2 & 3 \ 1 & 2 \end{bmatrix}\]provides a good example because its determinant \( 2 \times 2 - 1 \times 3 = 1 \) is non-zero. This ensures that the matrix is invertible and thus suitable for encoding messages.

Matrix Multiplication

Matrix multiplication is the process used to encode messages in cryptography. It involves multiplying the coding matrix by a block of numbers representing the message. This multiplication transforms the original message into a cryptogram — a disguised form that’s not directly readable.
For example, if we encode the word 'RUN', which corresponds numerically to \( R = 18, U = 21, N = 14 \), the encoding involves multiplying these numbers with our coding matrix.

• Divide the message into pairs of numbers: \( (18, 21), (14, 0) \)
• Multiply each pair by the coding matrix \( A \)

This action results in a new set of numbers, the cryptogram, which represents the encoded message.

Cryptogram

A cryptogram is the result of encoding a message using matrix multiplication and a coding matrix. It's essentially a sequence of numbers that stand in for a word or a phrase. Since these numbers don't directly translate back into readable characters without the key (here, the inverse of the coding matrix), they offer a secure way to hide information.
The encoding process converts the message using the chosen coding matrix, resulting in the cryptogram. Continuing with our example of 'RUN', once the numerical representations of \( R, U, \) and \( N \) have been multiplied by matrix \( A \), the numbers produced give us the cryptogram that represents 'RUN' in a scrambled format.

Matrix Inverse

The matrix inverse is crucial for decoding the cryptogram back into the original message. If you have used a matrix to encode, then applying the inverse of that matrix will allow you to revert to the original data.
Finding the inverse involves a little algebra. For a 2x2 matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the inverse is given by\[A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}\]In our exercise, using the inverse matrix,\[A^{-1} = \begin{bmatrix} 2 & -3 \ -1 & 2 \end{bmatrix}\]to multiply the cryptogram will decode it back to the original numerical sequence of the message. This ensures that the original message can be accurately recovered, validating both the encoding and decoding process.