[link uses the term “key” where this blog distinguishes “password” and “key” as concepts]
A creative use for matrix multiplication is to multiply the matrix built from two comparable texts [two ciphertexts, as in a chosen plaintext attack, or plaintext-ciphertext, as in an adaptive chosen plaintext attack,] with its transpose matrix.
One early experiment would be to count related occurrences of a given character, and name rows and columns after the text character that mapped to it. Another would be to correlate character pairings, in a similar fashion to an attack on Playfair.
The [square] matrix of A x A^T would show one characteristic, but A^T x A would show a different quality. They are reciprocal, but depending on the function (cell contents) the meaning of the information contained in each might differ.