Finding inverse linear transformation I'm solving a homework question and I'm stuck with it's last part.
The question goes like this:

Let $\displaystyle T:M_n(\mathbb{R})\to M_n(\mathbb{R})$ be a transformation defined as $T(A)=A+2A^T$.

*

*Show that T is a linear transformation.

*Show that T is bijective.

*Find $T^{-1}$.


I solved parts $(1)$ and $(2)$ quite easily, but couldn't solve part $(3)$. I tried to look for some methods or solving it by trial and error, but all failed.
I will appreciate any kind of help, thank you!

Question 1 is just computations. For question 2 you want to show that $T$ is injective, which will suffice; so suppose $T(A)=0$, or
$$
A=-2A^T
$$
Transposing we get $A^T=-2A$, so from the other equation we have $A=4A$ that implies $A=0$.
The same idea can be used for finding the inverse: you want a solution for $B=A+2A^T$. Transposing we have
$$
\begin{cases}
B=A+2A^T\\
B^T=2A+A^T
\end{cases}
$$
Now we can eliminate $A^T$: multiply the second equation by $2$ and subtract the first equation:
$$
4A+2A^T-(A+2A^T)=2B^T-B
$$
so
$$
A=-\frac{1}{3}B+\frac{2}{3}B^T.
$$
Thus the inverse is the map 
$$
T^{-1}\colon X\mapsto -\frac{1}{3}X+\frac{2}{3}X^T
$$
