How would be the Diagonalizability ($A$),Eigenvalues($A$) and $det(A)$ are related to Diagonalizability ($T$),Eigenvalues($T$) and $det(T)$?

Let $ M_n(\mathbb R)$ denotes the set of $ n\times n$ real matrices. Let $ A\in M_n(\mathbb R)$ and let $ T:M_n(\mathbb R)\to M_n(\mathbb R)$ be a linear transformation defined by $ T(X)=AX, X\in M_n(\mathbb R).$ Then How would be the Diagonalizability ($ A$ ),Eigenvalues($ A$ ) and $ det(A)$ are related to Diagonalizability ($ T$ ),Eigenvalues($ T$ ) and $ det(T)$ ?

My Attempt:- Consider the equation $ T(X)=\lambda X\implies AX=\lambda X=(A-\lambda I_n)X.$ How do I find the relation among the diagonalizablity of $ A$ and $ T$ . I know that If $ T$ Linear transformation is diagonalizable. then $ \dim$ (Eigenspace($ \lambda$ ))=Muliplicity of ($ \lambda$ ), $ \forall \lambda $ Spectrum($ T$ ). Please help me.

I know the fact That $ rank(T)=n. rank(A)$ and $ Nullity(T)=n. Nullity(A)$