Q9. - Algebra 2

Question

Select a headline from a newspaper or the title of a magazine article and code it using your own coding matrix. Give your message and the coding matrix to a friend to decode. (Hint: Use a coding matrix whose determinant is 1 and that has all positive elements.)

Step-by-Step Solution

Answer

The coded message is: $\mathbf{13}\mathbf{\left|}\mathbf{9}\mathbf{\right|}\mathbf{32}\mathbf{\left|}\mathbf{20}\mathbf{\right|}\mathbf{18}\mathbf{\left|}\mathbf{9}\mathbf{\right|}\mathbf{24}\mathbf{\left|}\mathbf{21}\mathbf{\right|}\mathbf{31}\mathbf{\left|}\mathbf{22}\mathbf{\right|}\mathbf{29}\mathbf{|}\mathbf{24}$

Step 1 ­- Description of step.

Consider the coding matrix be $C=\left[\begin{array}{cc}2& 1\\ 1& 1\\ & \end{array}\right]$.

The value of the determinant of the coding matrix $C$ is:

$\begin{array}{c}\left|C\right|=\left|\begin{array}{cc}2& 1\\ 1& 1\end{array}\right|\\ =\left(2\right)\left(1\right)-\left(1\right)\left(1\right)\\ =2-1\\ =1\end{array}$

Therefore, it can be noticed that the value of the determinant of the coding matrix $C$ is 1 and the coding matrix $C$ has all positive elements.

Step 2 ­- Description of step.

Consider the selected title of a magazine article as DELHI_CRIMES.

Now convert the letters to numbers by using the codes.

Therefore, it can be obtained that:

$DELHI\_CRIMES\cong 4\left|5\right|12\left|8\right|9\left|0\right|3\left|18\right|9\left|13\right|5|19$

Step 3 ­- Write the message in matrix form.

The message matrix $B$ is given by:

$B=\left[\begin{array}{cc}4& 5\\ 12& 8\\ 9& 0\\ 3& 18\\ 9& 13\\ 5& 19\\ & \end{array}\right]$

Step 4 ­- Description of step.

Now multiply the message matrix by the coding matrix to get the coded message.

Therefore, it can be obtained that:

$\begin{array}{c}B\cdot C=\left[\begin{array}{cc}4& 5\\ 12& 8\\ 9& 0\\ 3& 18\\ 9& 13\\ 5& 19\end{array}\right]\cdot \left[\begin{array}{cc}2& 1\\ 1& 1\end{array}\right]\\ =\left[\begin{array}{cc}\left(4\right)\left(2\right)+\left(5\right)\left(1\right)& \left(4\right)\left(1\right)+\left(5\right)\left(1\right)\\ \left(12\right)\left(2\right)+\left(8\right)\left(1\right)& \left(12\right)\left(1\right)+\left(8\right)\left(1\right)\\ \left(9\right)\left(2\right)+\left(0\right)\left(1\right)& \left(9\right)\left(1\right)+\left(0\right)\left(1\right)\\ \left(3\right)\left(2\right)+\left(18\right)\left(1\right)& \left(3\right)\left(1\right)+\left(18\right)\left(1\right)\\ \left(9\right)\left(2\right)+\left(13\right)\left(1\right)& \left(9\right)\left(1\right)+\left(13\right)\left(1\right)\\ \left(5\right)\left(2\right)+\left(19\right)\left(1\right)& \left(5\right)\left(1\right)+\left(19\right)\left(1\right)\end{array}\right]\\ =\left[\begin{array}{cc}8+5& 4+5\\ 24+8& 12+8\\ 18+0& 9+0\\ 6+18& 3+18\\ 18+13& 9+13\\ 10+19& 5+19\end{array}\right]\\ =\left[\begin{array}{cc}13& 9\\ 32& 20\\ 18& 9\\ 24& 21\\ 31& 22\\ 29& 24\end{array}\right]\end{array}$

Step 5 ­- Description of step.

Therefore, the coded message is: $13\left|9\right|32\left|20\right|18\left|9\right|24\left|21\right|31\left|22\right|29|24$.