A probability problem by XY Y

Using a digital computer, try to invert all $26^4=456976$ matrices to see which matrices were successfully inverted.

使用數字計算機，嘗試反轉所有 $26^4=456976$ 矩陣，以查看哪些矩陣成功反轉。

The inverse of $\left( \begin{array}{cc} a & b \\ c & d \\ \end{array} \right)$ is $\frac{\left( \begin{array}{cc} d & -b \\ -c & a \\ \end{array} \right)}{a d-b c}$ .

Only the integers 1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23 and 25 have multiplicative inverses in $\mathbb{Z}_{26}$ . Therefore the only matrices that can be inverted are those that are non-singular in the normal matrix sense and for which the fraction $\frac{1}{a d-b c}$ can be inverted to a multiplication by having a denominator that is one of those previously mentioned integers with multiplicative inverse in $\mathbb{Z}_{26}$ .

In $\mathbb{Z}_{26}$ , the multiplicative inverses are, respectively: $1\to 1,\,3\to 9,\,5\to 21,\,7\to 15,\,9\to 3,\,11\to 19,\,15\to 7,\,17\to 23,\,19\to 11,\,21\to 5,\,23\to 17,$ and $25\to 25$ .