Q39. - Algebra 2

Question

For Exercises 39-41, use the alphabet listed below.

Your friend has sent you a series of messages that were coded with the coding matrix $\mathbf{C}\mathbf{=}\left[\begin{array}{cc}2& 1\\ 1& 1\end{array}\right]$. Use the inverse of matrix to decode each message.

39. $\mathbf{50}\mathbf{\left|}\mathbf{36}\mathbf{\right|}\mathbf{51}\mathbf{\left|}\mathbf{29}\mathbf{\right|}\mathbf{18}\mathbf{\left|}\mathbf{18}\mathbf{\right|}\mathbf{26}\mathbf{\left|}\mathbf{13}\mathbf{\right|}\mathbf{33}\mathbf{\left|}\mathbf{26}\mathbf{\right|}\mathbf{44}\mathbf{\left|}\mathbf{22}\mathbf{\right|}\mathbf{48}\mathbf{\left|}\mathbf{33}\mathbf{\right|}\mathbf{59}\mathbf{\left|}\mathbf{34}\mathbf{\right|}\mathbf{61}\mathbf{\left|}\mathbf{35}\mathbf{\right|}\mathbf{4}\mathbf{|}\mathbf{2}$

Step-by-Step Solution

Answer

The decoded message is: $\mathbf{NVVG}\mathbf{\_}\mathbf{RM}\mathbf{\_}\mathbf{GSV}\mathbf{\_}\mathbf{ORYIZIB}\mathbf{\_}$

Step 1: Identify the coding matrix and coded message

The coded message is given as a sequence of numbers. The coding matrix \(C\) is provided, and we need to find the inverse matrix \(C^{-1}\) to decode.

Step 2: Find the inverse of the coding matrix

Compute \(C^{-1}\) using the formula for the inverse of a 2×2 matrix (or appropriate method for larger matrices). Multiply each block of the coded message by \(C^{-1}\).

Step 3: Decode the message

Multiply the coded message blocks by \(C^{-1}\) to obtain numerical values, then convert these numbers to letters using the provided alphabet key to reveal the decoded message.