Mathick
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Suppose that we are in the situation that Alice is using a Hill cipher consisting of a $2 \times 2$ matrix $M$ to send her message, which is $100$ ‘A’s. If Eve intercepts this message and knows that plaintext contained only one letter, and she also knows anyone of the entries of the matrix $M$, then prove that Eve can use this information to find the plaintext and the complete key.
So I tried writing a matrix $M$ as $M = \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix}$ and assuming that Eve saw messages $c_1$ and $c_2$, I got
$\begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} \begin{bmatrix} x \\ x \\ \end{bmatrix} = \begin{bmatrix} c_1 \\ c_2 \\ \end{bmatrix}$ which implies $x \begin{bmatrix} a + b \\ c + d \\ \end{bmatrix} = \begin{bmatrix} c_1 \\ c_2 \\ \end{bmatrix}$ where $x$ is a letter that Eve wants to find.
But I don't know how to conclude from this that she can do it. I tried assuming that, for example, she knows the entry $a$ or any other and proceed but it led me nowhere.If you could help or give me some hint, I would appreciate it very much.
So I tried writing a matrix $M$ as $M = \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix}$ and assuming that Eve saw messages $c_1$ and $c_2$, I got
$\begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} \begin{bmatrix} x \\ x \\ \end{bmatrix} = \begin{bmatrix} c_1 \\ c_2 \\ \end{bmatrix}$ which implies $x \begin{bmatrix} a + b \\ c + d \\ \end{bmatrix} = \begin{bmatrix} c_1 \\ c_2 \\ \end{bmatrix}$ where $x$ is a letter that Eve wants to find.
But I don't know how to conclude from this that she can do it. I tried assuming that, for example, she knows the entry $a$ or any other and proceed but it led me nowhere.If you could help or give me some hint, I would appreciate it very much.