So I assume the input numbers are from 1 to 26 where A=1, B=2 etc as is usual for these types of codes.

I let the encoding matrix (E) be:

\(\displaystyle \left( \begin{smallmatrix} a&b\\ c&d \end{smallmatrix} \right)\)

Assuming ad - bc =1 (for now - we can adjust that later by using a multiplier out the front of the matrix if we need to). This will also help with the condition that the decoding matrix has integers (if the encoding one does).

Because your output numbers (without using modulo arithmetic) are not too large and these are generated by using multiples of 1-26 added together

I started off setting a=2 (smallest even number).

So E = \(\displaystyle \left( \begin{smallmatrix} 2&b\\ c&d \end{smallmatrix} \right)\)

The decoding matrix D = \(\displaystyle \left( \begin{smallmatrix} d&-b\\ -c&a \end{smallmatrix} \right)\)

Now since "Position 1,1 in the encoding matrix is the negative of the number in position 1,1 in the decoding matrix", then d=-2

So E = \(\displaystyle \left( \begin{smallmatrix} 2&b\\ c&-2 \end{smallmatrix} \right)\)

For the det to be equal to 1 ie -4 - bc =1, bc=-5. Keeping to integers there are a limited number of combinations,

b=1 and c=-5, b=-1 and c=5, b=5 and c=-1, b=-5 and c=1

So D could equal =\(\displaystyle \left( \begin{smallmatrix} -2&1\\ -5&2 \end{smallmatrix} \right)\) or one of the other 3 combinations (or a multiple thereof)

If you multiply this matrix by (91, -38) {written as a column}you should expect output between 1 and 26, but you don't get that. Since modulo arithmetic is not used, then that wasn't the correct D.

So try another combination. Eventually one of them worked (YAY!!) and I successfully decoded the word. Have a go and let me know what you get.