Let $M_2(\mathbb R)$ denotes the set of $2\times2$ real matrices. Let $A\in M_2(\mathbb R)$ be of trace $2$ and determinant $-3$. Identifying $M_2(\mathbb R)$ with $\mathbb R^4$, consider the linear transformation $T:M_2(\mathbb R) \to M_2(\mathbb R): B \mapsto AB$. Then which of the followings are true:
(1) $T$ is diagonalizable,
(2) $2$ is an eigenvalue of $T$,
(3) $T$ is invertible,
(4) $T(B)=B$ for some $0\neq B \in M_2(\mathbb R)$.
Here's how I tried it: Since $0$ is not an eigen value of $A$ so $T$ is so option (3) is correct. To show (2) & (4) are incorrect I considered the matrix $ \begin{pmatrix} 3&0\\ 0&-1\end{pmatrix}\quad $ which satisfies all the conditions of $A$ & noticed that both $T(B)=2B$ & $T(B)=B$ yeild $B=0$.But I'm clueless about the option (1). Please help.