I am trying to better understand cryptography using Matrices. I have the following Ciphertext which was encoded using the key matrix $$A=\begin{bmatrix}3&-1&2\\4&1&0\\5&0&3\end{bmatrix}$$
Ciphertext: ZAQUQTBLSTRTCICOMAPEIRXB
I turned this into the matrix $$C=\begin{bmatrix}25&20&1&19&2&14&15&17\\0&16&11&17&8&12&4&23\\16&19&18&19&2&0&8&1\end{bmatrix}$$
I have: $$A^{-1}=\frac{1}{11} \begin{bmatrix}3&3&-2\\-12&-1&8\\-5&-5&7\end{bmatrix}$$ I looked at an example on Wikipedia (under the "decrypt" section). I need to change this to modulo 26, so was thinking that I use $$(A^*)^{-1}=\begin{bmatrix}3&3&24\\14&25&8\\21&21&7\end{bmatrix}$$
Unfortunately, $(A^*)^{-1}C$ does not give me a coherent message and $A^{-1}C$ doesn't even give me whole numbers.
I know there is a problem with setting up C- I start with the letter Z, but I can't be sure if the first entry of the matrix should be 25 or 51,-1,77 etc.
What should I do next?
Any help would be greatly appreciated!
Assuming the $\frac{1}{11}$ is correct, you should replace it by $19$, as $11 \times 19 = 209 \equiv 1 \pmod{26}$ so that is the inverse of $11$ modulo $26$. So all entries get multiplied by $19$ so $3$ becomes $3 \times 19 \equiv 7$ etc.
Recompute with the new matrix.