Solution of Problem 1

Given $A = \left[ \begin{array}{rr} 4&1\\ 3&2\\ \end{array}\right]$, we can find

$$A^{-1} = {( 8-3) }^{-1} \left[ \begin{array}{rr} 2&-1\\ -3&4... ...-1} \left[ \begin{array}{rr} 2&-1\\ -3&4\\ \end{array}\right]$$

But this matrix does not have integer entries, and is not very halpful. Since oure conversion table has 26 letters, we may want to use modular arithmetic, with $\mbox {mod}26$. Since $5 \times 21 = 1 \mbox {mod}26$, we may replace $\frac{ 1} {5}$ by 21. Therefore,

$$A^{-1} = 21 \left[ \begin{array}{rr} 2&-1\\ -3&4\\ \end{arr... ...t] =\left[ \begin{array}{rr} 16&5\\ 15&6\\ \end{array}\right]$$

in the multiplications above modular arithmetics ( $\mbox {mod}26$) is used.

To double check, find $A A^{-1}$,which should be equal to the identy matrix.

$$\left[ \begin{array}{rr} 4&1\\ 3&2\\ \end{array}\right] \l... ...\left[ \begin{array}{rr} 1&0\\ 0&1\\ \end{array}\right] = I.$$

Now, knowing that $A^{-1} =\left[ \begin{array}{rr} 16&5\\ 15&6\\ \end{array}\right]$, we can divide the whole message into pairs, and convert them into their correspoing numerical values.

11 14 15 24 1 15 10 24

Now multiply $A^{-1}$ by each of these encoded vector, the numbers change to

$$246 249 360 369 91 105 280 294$$

Using modular arithmetic, mod 26, the numbers become

12 15 22 5 13 1 20 8

Which converts to the following string of letters:

L O V E M A T H

This yields the message,

LOVE MATH