Is $ f(A) = A + 2A^{T} $ an isomorphism of $ \mathbb R^{5,5} $ onto itself?

Hint:

Check if the kernel of this transformation is trivial. This is enough to establish if it is isomorphism: $$\begin{eqnarray} f(A) =0 &\implies &A=-2A^T \\ &\implies &A^T=(-2A^T)^T \\ &\implies &A^T = -2A \\ &\implies &A =4A \\ &\implies &A=0 \end{eqnarray} $$

You have $A-B = 2(A-B)^\top$. This implies $a_{ij} - b_{ij} = 2 (a_{ji} - b_{ji}) = 4(a_{ij} - b_{ij})$, so...