$(A:M_{2}->M_{2})$ - multiplying right-side on matrix (1). Is this a unitary operator? Dot product is $(A,B) = \sum_{i=1}^{2}\sum_{j=1}^{2}a_{ij}b_{ij}$.
$ \bigl(\begin{smallmatrix} 13 & 46\\ 23/2 & 13 \end{smallmatrix}\bigr) $ (1)
And sorry for my English.
Certainly not - it is not even an isometry. Prove this by applying your operator to the matrix with all entries equal to $1$.