## 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 $$Nullity(T)=n. Nullity(A)$$