I'm creating a code that that uses a matrix or matrices as a key. For example, given each string of $n$ letters, construct it into a vector using its position in the alphabet, and multiply it by an $n\times{n}$ matrix and the resulting vector is the encoded string.
Such a matrix would have to be non-singular, and it would have to be invariant on the set of vectors such that all of its components are positive integers and none of them are greater than $26$ (or how many letters there are in the particular character set). Is such a matrix (besides the identity matrix and permutation matrices) possible?
I suspect that it is impossible because the vector set is not a vector space, nor is it a subspace, but I don't know how to justify this.
If it is impossible to find such a matrix, can you provide a proof to demonstrate it?
Here is an alternate take. Suppose you are encoding bits instead of the alphabets (you can convert an alphabet to its bit representation so this is not restrictive). The probability that a random $n\times n$ binary matrix is invertible (even over $\mathbb{F}_2$) is positive (around $.29$) as $n\rightarrow\infty$ (see e.g. Probability that a random binary matrix is invertible?). So you have a lot of chance in finding such matrices. Also, according to what Will Orrick says on the same page, if you allow inversions over $\mathbb{R}$ the probability that your random $0$-$1$ matrix will be non-singular as $n\rightarrow\infty$ according to Tao and Vu.