Let T : $M_n(\mathbb{R})$ to $M_n(\mathbb{R})$ be the linear transformation then what will be the determinant, trace, characteristic polynomial,eigen values, rank and minimal polynomial of following linear transformation:
$(1)$ $T(A)=BA $, for any fixed matrix $B\in M_n(\mathbb{R})$
$(2)$ $T(A)=AB$, for any fixed matrix $B\in M_n(\mathbb{R})$
$(3)$ $T(A)=BA+AB$, for any fixed matrix $B\in M_n(\mathbb{R})$
$(4)$ $T(A)=BA-AB$, for any fixed matrix $B\in M_n(\mathbb{R})$
(5) $T(A)=AB-BA$, for any fixed matrix $B\in M_n(\mathbb{R})$
I tried, for particular case $n=2$ the determinant and trace of $(1),(2)$ is $\operatorname{trace} T=2(\operatorname{trace} B)$ and $ \det T=(\det B)^2$. Is it hold for any $n\in \mathbb{N}$?
For $(3)$ $\operatorname{trace} T=4(\operatorname{trace} B)$ and $ \det T=4(\det B)(\det B)^2$ In general, is it true that $\operatorname{trace} T=n^2(\operatorname{trace} B)$? and what about determinant?
For $(4),(5)$ trace and determinant is zero. Once I know the determinant I can find relation between eigen values of A and B. I know idea about minimal polynomial. If the question partially discussed in this site one can send the link on comment box. I want to know that the thing holds in particular case of $n$ for $T$, Is it hold in general? Please help me. Thanks in advance.