Matrices are commonly used to encrypt data. Here is a simple form such an encryption can take. First, we represent each letter in the alphabet by a number, so let us take $\left[\begin{array}{ccccccccccccc}1& 2& 15& 18& 20& 0& 13& 9& 19& 19& 9& 15& 14\end{array}\right].A22\left[\begin{array}{cc}1& 3\\ 4& 2\end{array}\right]$A Encrypted Matrix $=\left[\begin{array}{ccc}1& 3& 2\end{array}\right]$

$4\left[\begin{array}{ccccccccccccc}1& 15& 20& 13& 19& 9& 14& 18& 0& 9& 19& 15& 0\end{array}\right]$

$2$

$=\left[\begin{array}{ccccccccccccc}7& 69& 20& 40& 76& 54& 14& 96& 80& 70& 114& 66& 56\end{array}\right]$

$8\left[\begin{array}{cccccccccccccc}7& 8& 69& 96& 20& 80& 40& 70& 76& 114& 54& 66& 14& 56\end{array}\right].A^{-1}AA\left[\begin{array}{cccccccccccccc}48& 42& 72& 108& 14& 26& 20& 80& 43& 32& 88& 122& 59& 56\end{array}\right]0,A=1,B=2,$ and $soon.$ Thus, for example, "ABORT MISSION" becomes

$\left[\begin{array}{ccccccccccccc}1& 2& 15& 18& 20& 0& 13& 9& 19& 19& 9& 15& 14\end{array}\right].$

$To$ encrypt this coded phrase, $we$ use $an$ invertible matrix $of$ any size with integer entries. For instance, let $us$ take $Atobe$ the $22$ matrix $\left[\begin{array}{cc}1& 3\\ 4& 2\end{array}\right].We$ can first arrange the coded sequence $of$ numbers $in$ the form $of$ a matrix with two rows $(using$ zero $in$ the last place $ifwe$ have $an$ odd number $of$ characters$)$ and then multiply $on$ the left $byA.$

Encrypted Matrix $=\left[\begin{array}{ccc}1& 3& 2\end{array}\right]$

$4\left[\begin{array}{ccccccccccccc}1& 15& 20& 13& 19& 9& 14& 18& 0& 9& 19& 15& 0\end{array}\right]$

$2$

$=\left[\begin{array}{ccccccccccccc}7& 69& 20& 40& 76& 54& 14& 96& 80& 70& 114& 66& 56\end{array}\right]$

$8$

which $we$ can also write $as$

$\left[\begin{array}{cccccccccccccc}7& 8& 69& 96& 20& 80& 40& 70& 76& 114& 54& 66& 14& 56\end{array}\right].$

$To$ decipher the encoded message, multiply the encrypted matrix $by{A}^{-1}.$ The following question uses the above matrix A for encoding and decoding.

Decode the following message, which was encrypted using the matrix $A.(Include$ any appropriate spaces $in$ your answer.$)$

$\left[\begin{array}{cccccccccccccc}48& 42& 72& 108& 14& 26& 20& 80& 43& 32& 88& 122& 59& 56\end{array}\right]$