A transformation $T(x, y) = (ax + by + e, cx + dy + f)$ is an isometry. If it maps the origin $(0,0)$ to $(3,4)$ and the vector $(1,0)$ to a point that maintains a distance of $1$ from $(3,4)$, what can we conclude about the matrix $M = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$?