$\backslash$table$[[$A$,$B$,$C$,$D$,$E$,$F$,$G$,$H$,$I$],[10,21,4,15,11,5,0,20,14],[$J$,$K$,$L$,$M$,$N$,0,$P$,$Q$,$R$],[9,25,2,18,26,1,22,13,7],[$S$,$T$,$U$,$V$,$W$,$x$,$Y$,$Z$,$Blank$],[8,17,24,16,23,3,6,19,12]]$

The matrix A shown below will be used as the encoding matrix:

$A=\left[\begin{array}{ccc}2& 1& 4\\ 3& 2& 5\\ 1& 0& 2\end{array}\right]$

Since this is a $3x3$ matrix, the information must be entered in sets of three characters at a time.

Instructions

Show and explain all steps in your responses to the following:

Part $1$: Convert the statement 'MATH IS COOL' into a numerical sequence.

Part $2$: Use the matrix $A$ to encode the message 'MATH IS COOL'.

Part $3$: Find $A^{-1}.$

Part $4$: Calculate $A^{-1}A.$

Part $5$: Use $A^{-1}$ on your answer from part $3$ to decode the message.Scenario

Behind the scenes of our everyday life, systems of equations are being applied all the time. A system of equations is used any time information needs to be encoded or decoded, so virtually all information transferred on the Internet undergoes some level of encoding. For sensitive information, like credit card numbers or personal data, this encryption is vital. The way encoding works is input data is received and used as the initial values to enter into a system of equations. After computing the results, the output is an encoded message that can be decoded only by knowing the system of equations used.

When we write a system of equations as a matrix, it acts in the same way a decoder ring would by telling us how to convert our input data into an encrypted message. Then, the inverse of the encoding matrix is used for decoding the encrypted message.

The following table will be used to convert a written message into numeric data: