Let $T_1 : \mathbb{R}^{3 \times 3} \to \mathbb{R}$ and $T_2 : \mathbb{R}^{3 \times 3} \to \mathbb{R}^{3 \times 3}$ are linear transformation, where $T_2(A) = A^T$. If $ A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}$ and ($T_1 \circ T_2)(A) = T_1(A)$.
Find $T_1(A)$
What I'm doing is to write
$\begin{align} (T_1 \circ T_2)(A) &= T_1(T_2(A)) \\ T_1 (A)&= T_1(A^T) \end{align}$.
From this result I'm sure that I can't say $A=A^T$. Because I have no infomation that $T_1$ is injective. But $T_1$ has to be a linear transformation that brings every matrix in $\mathbb{R}^{3 \times 3}$ to $\mathbb{R}$. What I have to do next?