Chapter 8.5, Problem 57E

To determine

The decoding of cryptogram using ${\text{A}}^{\text{-1}}$

Answer to Problem 57E

The numbers are converted to the alphabets they are assigned to we get the message

  $\text{HAPPY}\text{NEW}\text{YEAR}$

Explanation of Solution

Given information:

We have to decode the following cryptogram:

  $11216411225502953234640755592$

Formula used:

Matrix multiplication is used.

Calculation:

To decode the following cryptogram:

  $11216411225502953234640755592$

Using $A=\left[\begin{array}{cc}1& 2\\ 3& 5\end{array}\right]$

Dividing them into $1\times 2$ matrix as below,

  $\left[1121\right]\left[64112\right]\left[2550\right]\left[2953\right]\left[2346\right]\left[4075\right]\left[5592\right]$

These have to be decoded using $A^{-1}$

  $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ , Then $A^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$

Therefore,

  $\begin{array}{l}{A}^{-1}=\frac{1}{\left(1\times 5\right)-\left(2\times 3\right)}\left[\begin{array}{cc}5& -2\\ -3& 1\end{array}\right]\\ =\frac{1}{5-6}\left[\begin{array}{cc}5& -2\\ -3& 1\end{array}\right]\\ =-1\left[\begin{array}{cc}5& -2\\ -3& 1\end{array}\right]\\ A^{-1}=\left[\begin{array}{cc}-5& 2\\ 3& -1\end{array}\right]\end{array}$

Now to get the uncoded row matrices,

Coded matrix Decoding matrix Uncoded Matrix

  $\begin{array}{l}\left[\begin{array}{cc}11& 21\end{array}\right]\left[\begin{array}{cc}-5& 2\\ 3& -1\end{array}\right]\left[\begin{array}{cc}8& 1\end{array}\right]\\ \\ \left[\begin{array}{cc}64& 112\end{array}\right]\left[\begin{array}{cc}-5& 2\\ 3& -1\end{array}\right]\left[\begin{array}{cc}16& 16\end{array}\right]\\ \\ \left[\begin{array}{cc}25& 50\end{array}\right]\left[\begin{array}{cc}-5& 2\\ 3& -1\end{array}\right]\left[\begin{array}{cc}25& 0\end{array}\right]\\ \\ \left[\begin{array}{cc}29& 53\end{array}\right]\left[\begin{array}{cc}-5& 2\\ 3& -1\end{array}\right]\left[\begin{array}{cc}14& 5\end{array}\right]\\ \\ \left[\begin{array}{cc}23& 46\end{array}\right]\left[\begin{array}{cc}-5& 2\\ 3& -1\end{array}\right]\left[\begin{array}{cc}23& 0\end{array}\right]\\ \\ \left[\begin{array}{cc}40& 75\end{array}\right]\left[\begin{array}{cc}-5& 2\\ 3& -1\end{array}\right]\left[\begin{array}{cc}25& 5\end{array}\right]\\ \\ \left[\begin{array}{cc}55& 92\end{array}\right]\left[\begin{array}{cc}-5& 2\\ 3& -1\end{array}\right]\left[\begin{array}{cc}1& 18\end{array}\right]\end{array}$

The following uncoded row matrices,

  $\left[\begin{array}{cc}8& 1\end{array}\right]\left[\begin{array}{cc}16& 16\end{array}\right]\left[\begin{array}{cc}25& 0\end{array}\right]\left[\begin{array}{cc}14& 5\end{array}\right]\left[\begin{array}{cc}23& 0\end{array}\right]\left[\begin{array}{cc}25& 5\end{array}\right]\left[\begin{array}{cc}1& 18\end{array}\right]$

Splitting them we get,

8 1 16 16 25 0 14 5 23 0 25 5 1 18

When the numbers are converted to the alphabets they are assigned to we get the message

  $\text{HAPPY}\text{NEW}\text{YEAR}$

Conclusion:

The numbers are converted to the alphabets they are assigned to we get the message

  $\text{HAPPY}\text{NEW}\text{YEAR}$