Summary: Domain and codomain are square matrices. Domain and codomain both have dimension 4. Not injective, not surjective, not invertible, 3 distinct eigenvalues, diagonalizable.
Definition:\begin{align*}\ltdefn{T}{M_{22}}{M_{22}},\quad \lt{T}{\begin{bmatrix}a&b\\c&d\end{bmatrix}}= \begin{bmatrix} -2a+15b+3c+27d & 10b+6c+18d \\ a-5b -9d & -a-4b-5c-8d \end{bmatrix} \end{align*}