Cryptography Application

Linear Algebra (Matrices) · Axiom Academy

A secret message, locked by a matrix and a determinant — and only one key matrix can ever open it again. You want to send MATH so that only the holder of a secret matrix can read it. In 1929 Lester Hill turned that idea into linear algebra: scramble the letters with a matrix, then undo the scramble with its inverse — and a determinant decides whether the lock can ever be opened. Lock the message with a matrix Letters become numbers ( ). Each pair of letters is a vector , and the key matrix encrypts it with . Type a word and watch each block get scrambled. An odd-length message is padded with a trailing X so it fills whole blocks of 2. Open it back up with the inverse To read the message you run the scramble backward: . The inverse key mod 26 is , because . Decrypt the ciphertext from Beat 1 and watch the original word reappear. Beat 1's result is dropped in here automatically — or type any ciphertext. Which keys can actually be opened? Not every matrix works. An inverse mod 26 exists only when is invertible mod 26 — that is, and . Pick a key and see whether it locks a message you could ever unlock again. The whole cipher rides on one determinant : must be nonzero and coprime to 26 , or no inverse exists and the message is lost. That is the real-world face of this unit — a determinant tells you whether a matrix transform can be undone , and an inverse is the tool that undoes it. The same test decides reversibility in computer graphics, error-correcting codes, and solving .

This is the written version of the interactive lesson above. See the full Linear Algebra (Matrices) course.