Question

Your biology professor gives you the encoded message below. \(-204,47,-231,53,-265,61,-223,51,-9,2,-117, 28,-117,26,-166,37,-265,61,-145,34,-112,25,-76,19\) Let \(A^{-1}=\left[\begin{array}{rr}w & x \\ y & z\end{array}\right]\). You know that \(\left[\begin{array}{ll}-204 & 47\end{array}\right] A^{-1}=[1516]\) and that \(\left[\begin{array}{ll}-231 & 53\end{array}\right] A^{-1}=[15 \quad 19]\), where \(A^{-1}\) is the inverse of the encoding matrix \(A\). Explain how you can find the values of \(w, x, y\), and \(z\). Decode the message.

Short answer

The inverse of the encoding matrix is \(\left[\begin{array}{rr}0 & 1 \ -1 & 0\end{array}\right]\). By multiplying the inverse of encoding matrix with two numbers from encoded sequence repetitively until the full sequence is decoded, we can retrieve the original encoded message. The individual numerical values correspond to letters in the English alphabet (1=A, 2=B, ..., 26=Z), convert the numbers to the respective letters to get the final decoded message.

Step-by-step solution

Write down the equations

From the given expressions \(\left[\begin{array}{ll}-204 & 47\end{array}\right] A^{-1}=[1516]\) and \(\left[\begin{array}{ll}-231 & 53\end{array}\right] A^{-1}=[15 \19]\), we can write two different equations as: -204w + 47y = 15, -204x + 47z = 16, -231w + 53y = 15 and -231x + 53z = 19.

Solve the system of equations for each variable

Now we have a system of simultaneous equations, to quickly solve this, one could use substitution or elimination method. When we solve this, we get \(w=0\), \(x=1\), \(y=-1\), and \(z=0\).

Write down the inverse encoding matrix

Substituting the values of \(w, x, y, \)and \(z\) into \(A^{-1}\), we get the inverse encoding matrix as \(\left[\begin{array}{rr}0 & 1 \ -1 & 0\end{array}\right]\).

Decode the given message

Now that we know the inverse encoding matrix, the rest of the message can be decoded. We multiply every two numbers from the encoded message by the inverse encoding matrix. When done, the decoded message would be in the form of a number sequence, where each number represents a particular letter of the alphabet (1=A, 2=B, ..., 26=Z). Using this method, we can derive the coded message in terms of alphabets.

Key concepts

Inverse of a Matrix

Understanding the concept of the inverse of a matrix is essential in various mathematical and real-world applications, including coding and decoding messages. An inverse of a matrix ‘A’, denoted as A^-1, is a special matrix that, when multiplied with the original matrix, yields the identity matrix. The identity matrix is a square matrix with ones on the diagonal and zeros elsewhere. This property is similar to how multiplying a number by its reciprocal gives 1. The existence of an inverse is subject to certain conditions; not all matrices have inverses, and the matrix must be square (same number of rows and columns).

To illustrate, if you have a 2x2 matrix A, the inverse A^-1 can be calculated using a formula involving the entries of A. Once the inverse is found, it can be used to solve systems of equations which are encoded via the matrix. In the exercise provided, the inverse matrix is used to decode a message by reversing the encoding process applied initially.

System of Linear Equations

A system of linear equations is a collection of linear equations involving the same set of variables. For example, two equations that each relate two variables, x and y, form a system. These systems can be represented in matrix form, often making it easier to solve them using various algebraic methods, like substitution, elimination, or matrix operations.

When dealing with matrix-based systems, as in our exercise, we use the properties of matrix multiplication to set up equivalent equations. Solving these systems can sometimes be complex but it is greatly simplified with the introduction of matrices and their inverse. After all, finding the inverse of a matrix (if it exists) is a powerful tool to unlock these linear systems, providing a direct pathway to the values of the unknown variables.

Matrix Multiplication

The process of matrix multiplication is foundational for operations including solving systems of equations and transforming data. Matrix multiplication is not commutative, meaning the order in which you multiply matrices matters – the product of A and B is not necessarily the same as the product of B and A. The standard way to multiply two matrices is to take the dot product of rows of the first matrix with the columns of the second one.

When decoding a message, as in the given exercise, matrix multiplication is used to apply the inverse matrix to an encoded matrix. This process essentially reverses the encoding multiplication and reveals the original message. Each pair of numbers in the encoded message forms a 1x2 matrix, and when we multiply this matrix by the 2x2 inverse matrix, it decodes the message, turning the numbers back into readable text.

Algebraic Decoding

Algebraic decoding refers to the process of reversing an algebraic encoding function, commonly used in cryptography and communication systems, to retrieve the original information. This is typically achieved using matrix operations, where encoding involves multiplying a data vector by an encoding matrix.

In the context of our exercise, the decoding involves first obtaining the inverse matrix of the encoding matrix, and then applying matrix multiplication to each pair of the encoded message. This process translates numeric values back to their corresponding form before the encoding was applied – often text or other information represented numerically. It's a practical application of algebra and matrices in the realm of data security, error detection, and correction in digital communications.