We next enter the coded message into a new matrix $B,$ by entering it letter by

letter into the columns of $B.$ Note that since the encrypting matrix $A$ is a $33,B$

must have a $3$ rows for multiplication to be possible.

$B=\left[\begin{array}{ccccc}M& H& S& W& T\\ A& E& ?& ?& ?\\ T& \text{I}& S& E& ?\end{array}\right]=\left[\begin{array}{ccccc}7& -4& 10& 12& -10\\ 1& 0& 0& 3& 0\\ -10& 5& 10& 3& 0\end{array}\right]$

If the message is too short to fill matrix $B,$ we use blank spaces to complete the final

column.

Step $2$: Use the matrix $A$ to encrypt it$.$

Computing the product $AB$ encrypts the message, and only someone with access to

$A^{-1}$ will be able to read it$.$

$AB=\left[\begin{array}{ccccccc}-1& 5& -1& 11& 7& -5& 2\end{array}\right]$

$1$

$1$

$-2\left[\begin{array}{ccccccccccccc}7& -4& 10& 12& -10& 0& 0& 3& 0& 5& 10& 3& 0\end{array}\right]$

$-10$

$1$

$=\left[\begin{array}{ccccccccccccc}8& -1& -20& 0& 10& 43& 50& 30& 20& 6& 30& 3& -10\end{array}\right]$

$-18$

$-73$

The coded message is $8,-73,-18,-1,43,6,-20,50,30,0,30,3,10,20$ and

$-10.$

Deciphering a message

Step $3$: Decipher the coded message from step $2.$

The received message is $8,-73,-18,-1,43,6,-20,50,30,0,30,3,10,20$ and

$-10.$ It is the result of the product $AB.$ To find matrix $B,$ we apply $A^{-1}$ since

$A^{-1}(AB)=(A^{-1}A)B=B.$ Writing the received message in matrix form, we

have

$AB=\left[\begin{array}{ccccc}8& -1& -20& 0& 10\\ -73& 43& 50& 30& 20\\ -18& 6& 30& 3& -10\end{array}\right]$ Step $4$: Next multiply $AB$ by $A^{-1}$ on the left to determine matrix $B$

$A^{-1}(AB)=\left[\begin{array}{ccccccc}-57& 5& -46& 1& -9& 0& 1\end{array}\right]$

$1$

$1$

$-11\left[\begin{array}{ccccccccccccc}8& -1& -20& 0& 10& 43& 50& 30& 20& 6& 30& 3& -10\end{array}\right]$

$-18$

$-73$

$=\left[\begin{array}{ccccccccccccc}7& -4& 10& 12& -10& 0& 0& 3& 0& 5& 10& 3& 0\end{array}\right]$

$-10$

$1=B$

Writing matrix $B$ in sentence form gives $7,1,-10,-4,0,5,10,0,10,12,3,3,-10,$

$0$&$0$ and using the substitution cipher to replace numbers with letters,

reveals the message MATH IS SWEET.

Activity $1($Unlock a coded message$)$

Doctor Strange is in deep trouble. He is being pursed by Thanos and is presently hiding in a secret

location to escape from being captured by him. He needs to send messages to two of his comrades,

Thor and Captain America, to tell them about his situation and whereabouts. As such, he composes

his first message by replacing each letter in the message by a number according to the code in

Table $1$ below.

Table $1$: Coding system

As Doctor Strange is afraid that his message will be intercepted and cracked by Thanos, therefore,

he decides to send the message in code together with a super hi$-$tech LOCK to Thor and Captain

America to unlock the coded message.

LOCK$=\left[\begin{array}{ccc}1& 0& 1\\ -1& 1& 0\\ 0& 1& 2\end{array}\right]$

With the use of Table $1$ and LOCK, what is the message that Doctor Strange is trying to send to Thor

and Captain America? Please show your working clearly on how you unlock the code message.

$(40$ marks $)$ Activity $2($Design your own substitution cipher$)$

$($a$)$ Use the criteria indicated to create a pair of $33$ matrices $(D$&$D^{-1}),$ where the entries

of the matrix $D$ are all integers.

$(20$ marks $)$

$($i$)d_{2,1}=5$

$($ii$)d_{3,2}=-2$

$($b$)$ Create your own substitution cipher. Then use it and the matrix $D$ you created in $($a$)$ to

encrypt the title of your favourite movie.

$(20$ marks $)$

$($c$)$ Use the matrix $D^{-1}$ and the appropriate substitution cipher to decrypt the message from

$($b$).$

$(20$ marks $)$