Decrypting with an Inverse Matrix
Decrypt a coded message using an inverse matrix: multiply the inverse of the encoding matrix by the coded message matrix to recover the original numeric values representing letters.
Key Concepts
Decrypting a message encoded with matrix multiplication requires using the inverse of the original encoding matrix. Each encrypted matrix is multiplied by this inverse matrix. This calculation perfectly reverses the scrambling process, restoring the numbers to their simple substitution cipher form. These numbers can then be translated back into letters using the original cipher key.
To decrypt, multiply the encrypted matrix $[E]$ by the inverse of the encoding matrix $[A]$: $[E] \cdot [A]^{ 1}$. If $\begin{bmatrix} 38 & 50 \\ 22 & 31 \end{bmatrix}$ is encrypted, multiplying by $\begin{bmatrix} 1 & 1 \\ 2 & 3 \end{bmatrix}^{ 1}$ returns the original cipher matrix $\begin{bmatrix} 14 & 12 \\ 4 & 9 \end{bmatrix}$.
How do you unscramble the matrix encrypted message? You use the ultimate undo button: the inverse matrix! Multiplying the scrambled message by the inverse of the secret encoding grid perfectly reverses the encryption, giving you back the original numbers, which you can then translate to letters.
Common Questions
How does matrix encryption and decryption work?
A message is converted to numbers (A=1, B=2 etc.) and arranged in a matrix. An encoding matrix A multiplies the message matrix to produce the coded matrix. To decrypt, multiply the coded matrix by A-inverse, which reverses the encoding and recovers the original message numbers.
How do you find the inverse of a 2x2 encoding matrix?
For A=[[a,b],[c,d]], the inverse is (1/det(A))*[[d,-b],[-c,a]] where det(A)=ad-bc. The inverse only exists if det(A) is not zero. Multiply each entry of the adjugate matrix by 1 divided by the determinant to get the inverse.
Why must the encoding matrix have a nonzero determinant?
A matrix with determinant zero is singular and has no inverse. Without an inverse, decryption is impossible because the encoding transformation cannot be reversed. Choosing an encoding matrix with a nonzero determinant guarantees that every encoded message can be uniquely decrypted.