Problem 78 - 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. $$ \begin{array}{llllllllll} {A} & {R} & {T} & {-E} & {N} & {R} & {I} & {C} & {H} & {E} & {S} \end{array} $$$$ \begin{array}{ccccccccccc} {1} & {18} & {20} & {0} & {5} & {14} & {18} & {9} & {3} & {8} & {5} & {19} \end{array} $$

Step-by-Step Solution

Answer

The numeric representation of 'ART-ENRICHES' is encoded with a chosen 2x2 matrix. Then the inverse of this matrix is found and used to decode the cryptogram back into the original numeric representation of the word.

Step 1: Choose a coding matrix \(A\)

Let's choose a 2x2 coding matrix. We can assign arbitrary values for the chosen matrix since the exercise does not give specific instructions on what matrix to use. For this solution, let's use the matrix \[ A = \left[ \begin{array}{cc} 1 & 2 \ 3 & 4 \end{array} \right] \].

Step 2: Encode the message

Encode the word 'ART-ENRICHES' to the corresponding numbers 1, 18, 20, 0, 5, 14, 18, 9, 3, 8, 5, 19. Divide these numbers into groups of 2, since we will multiply them with a 2x2 matrix. Lastly, perform the matrix multiplication for each group by using the coding matrix \(A\). The result will represent the encoded message.

Step 3: Find the multiplicative inverse of matrix \(A\)

Next, find the inverse of the Matrix \(A\). This can be done by calculating the determinant of the matrix \(A\), namely \(1*4 - 2*3 = -2\), and applying the formula for the inverse of a 2x2 matrix, namely \[ A^{-1} = \frac{1}{-2} \left[ \begin{array}{cc} 4 & -2 \ -3 & 1 \end{array} \right] = \left[ \begin{array}{cc} -2 & 1 \ 1.5 & -0.5 \end{array} \right] \].

Step 4: Decode the cryptogram

Multiply the encoded cryptogram from Step 2 with the inverse matrix of \(A\). The result should correspond to the original number representation of the word 'ART-ENRICHES'.

Key Concepts

Coding Matrix

The coding matrix is the cornerstone of encrypting a message using matrix multiplication in cryptography. Its role is to transform plain text into encrypted text in a systematic way. The coding matrix, which is usually square (same number of rows and columns), should be chosen carefully to ensure that it is invertible; that is, a corresponding matrix inverse exists.

In our example, a 2x2 matrix is used. This size is chosen because it complements the letter-to-number conversion where characters are paired. Each number pair is multiplied by the coding matrix to give a new pair of numbers, constituting part of the encrypted message, or cryptogram. It's critical that this matrix is kept secret, as it is essentially the 'key' to the message. Without it, decoding the cryptogram would be nearly impossible for unintended receivers.

Matrix Inverse

The inverse of a matrix is comparable to the reciprocal of a number. If you multiply a matrix by its inverse, you get the identity matrix, just like if you multiply a number by its reciprocal, you get 1. This property is vital in cryptography for decoding messages. To find the inverse of a matrix, denoted as \(A^{-1}\), you typically use a mathematical procedure that involves the calculation of the matrix determinant and applying certain transformations.

For a 2x2 matrix like the coding matrix selected in our exercise, the inverse is calculated using a specific formula involving the elements of the matrix and its determinant. Finding the inverse is essentially 'unlocking' the coding matrix, which allows us to revert encrypted messages back to their original, understandable form. When the correct inverse is applied to an encrypted message, the original plaintext is revealed.

Cryptogram Encoding

Encoding a cryptogram is the process of converting plain text into encoded text using the coding matrix. In our example, each letter of the message 'ART-ENRICHES' is first converted to a numerical value, where A=1, B=2, ... , Z=26, and a space or hyphen can be designated as 0. These numbers are then organized into pairs and multiplied by the coding matrix.

This operation is a classic example of matrix multiplication, where rows of the first matrix (number pairs) are multiplied by the columns of the second matrix (coding matrix) to produce the encoded pairs. These new number pairs form the cryptogram, a jumble of numbers that seems random to anyone who doesn't have the key (coding matrix) to decode it.

Matrix Determinant

The determinant of a matrix is a special scalar value that provides important information about the matrix. For 2x2 matrices, the determinant is calculated using a simple formula: \(ad - bc\) for a matrix \(\left[ \begin{array}{cc} a & b \ c & d \end{array} \right]\). The determinant can tell us whether a matrix is invertible—the key characteristic needed for a matrix to serve as a coding matrix in cryptography.

If the determinant of a matrix is 0, the matrix cannot be inverted and, thus, is not suitable for encrypting messages. This is because there would be no way to decode the message afterwards. In our sample problem, the determinant is calculated to verify the invertibility of the coding matrix, ensuring that once a message is encoded, it can indeed be decoded later using the matrix inverse.