How Are Matrices Used to Code and Decode Messages?

Matrices code a message by multiplying numerical representations of letters by a coding matrix. The message is decoded by multiplying the coded values by the coding matrix's inverse matrix.

The Mechanism

This is HL-only content from AHL 1.14, Introduction to matrices. First, assign each character a number, such as A $=1$, B $=2$, ..., Z $=26$, and arrange the numbers into column vectors.

Suppose the message is “HI”, so H $=8$ and I $=9$. Its message vector is

$$\mathbf{x}=\begin{pmatrix}8\\9\end{pmatrix}.$$

For longer messages, split the numerical sequence into blocks containing two values because $A$ has two columns. Each block must keep the same order. If a final block is incomplete, the question must specify a padding value.

Choose the coding matrix

$$A=\begin{pmatrix}1&2\\1&3\end{pmatrix}.$$

The coded vector is

$$\mathbf{y}=A\mathbf{x}=\begin{pmatrix}1&2\\1&3\end{pmatrix}\begin{pmatrix}8\\9\end{pmatrix}=\begin{pmatrix}26\\35\end{pmatrix}.$$

Check the multiplication dimensions: a $2 \times 2$ matrix multiplied by a $2 \times 1$ vector produces a $2 \times 1$ coded vector. This dimension check catches a transposed message vector and prevents multiplying the matrices in the wrong order.

The matrix must be invertible, meaning $\det(A)\neq 0$. Here, $\det(A)=1$, so

$$A^{-1}=\begin{pmatrix}3&-2\\-1&1\end{pmatrix}.$$

For a general $2 \times 2$ matrix $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$, the inverse exists only when , and it reverses the transformation exactly. If the determinant is zero, different message vectors can map to the same coded vector, so unique decoding is impossible.

Decoding gives

$$A^{-1}\mathbf{y}=\begin{pmatrix}3&-2\\-1&1\end{pmatrix}\begin{pmatrix}26\\35\end{pmatrix}=\begin{pmatrix}8\\9\end{pmatrix},$$

which converts back to “HI”. The common misconception is that the coding matrix is used again to decode. This is incorrect: decoding requires $A^{-1}$, because $A^{-1}A=I$.

Exam Technique

For an HL short-response question, write the matrix equation before using your GDC. Check matrix dimensions and verify that the coding matrix is invertible; then convert the decoded numbers back into letters using the stated convention. On a GDC, enter the matrix and vector carefully, but show the defining matrix equation as working. Keep exact integer values where possible and state the final decoded text, not only the vector.