Matrices are a tool well-suited for cryptography because they can encipher arbitrarily large blocks of text in one operation. Typically, each block in the message is treated as a vector of bytes, meaning integers modulo 256.
When Sandra uses a matrix to encipher a message, Riva must use the inverse of that matrix to decipher the message. Let’s begin the discussion of matrix methods with a technique for inverting a matrix.
There are several ways of solving a matrix equation such as C = AP when there is known plaintext. Since Emily knows P and C, but does not know A, she can solve the equation for A by right-multiplying it by P' to get CP' = APP' = A. So Emily needs to invert P. Riva does the opposite. She knows A, but does not know P, so she needs to invert A. Left-multiplying the equation by A', she gets A'C = A'AP = P.