We will now decrypt the ciphertext "SYICHOLER" using the keyword "alphabet" and a 3x3 matrix. The first step is to create a matrix using the keyword (since the keyword is shorter than 9 letters, just start the alphabet again until the matrix is full).
Now we need to find the multiplicative inverse of the determinant (the number that relates directly to the numbers in the matrix. But first, to find the determinant, we need to evaluate the following algebraic expression.
Once we complete this step, we have to take the modulo 26 of that remaining number.
Now we can find the inverse of the determinant (11). But to do this we need to find the number between 1 and 25 that when multiplied by the determinant (11) equals 1 mod 26. The easiest to use is trial and error.
Once we eventually find that the inverse of the determinant modulo 26 is 19. We will need this number later, but meanwhile, we need to find the adjugate matrix (a matrix of the same size as the original), which requires the use of a lot of calculations using algebraic operations.
When we obtain the 9 numbers, we need to take the mod 26 of each of those answers.
Now we have our adjugate matrix. We need to multiply the inverse determinate (19) by each of the numbers in this new matrix. When we receive our results, we have to use modulo 26.
If we put all of this together, we get the expression:
Now we finally have the inverse key matrix. We have to now multiply by the first trigraph of the ciphertext "SYICHOLER".
Next, we need to multiply the inverse key matrix by the second trigraph.
Now the third and last trigraph:
If we combine our decrypted letters we get the new plaintext "we are safe".