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

Short answer

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

Step-by-step solution

­- Find the inverse of the coding matrix C.

It is being given that the coding matrix is $C=\left[\begin{array}{cc}2& 1\\ 1& 1\end{array}\right]$

The determinant of the coding matrix C is:$\begin{array}{c}\left|\begin{array}{cc}2& 1\\ 1& 1\end{array}\right|=\left(2\right)\left(1\right)-\left(1\right)\left(1\right)\\ =2-1\\ =1\end{array}$

As the determinant of the coding matrix is not equal to zero, therefore the inverse of the matrix C exist.

The inverse of $2\times 2$matrix$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ is given by$A^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$ and$ad-bc\ne 0$ .

Therefore, the inverse of the coding matrix C is given by:

$\begin{array}{l}\frac{1}{\left(2\right)\left(1\right)-\left(1\right)\left(1\right)}\left[\begin{array}{cc}1& -\left(1\right)\\ -\left(1\right)& 2\end{array}\right]=\frac{1}{2-1}\left[\begin{array}{cc}1& -1\\ -1& 2\end{array}\right]\\ =\frac{1}{1}\left[\begin{array}{cc}1& -1\\ -1& 2\end{array}\right]\\ =\left[\begin{array}{cc}1& -1\\ -1& 2\end{array}\right]\end{array}$