<s>[INST] determinant of the linear transformation $T(X) =\frac{1}{2} (AX+XA)$ Let $V$ vector space of all matrices $3\times3$, and let $A$ be the diagonal matrix :  
$$
\begin{pmatrix}
1 &  0 & 0\\
0  &  2& 0 \\
0  &  0& 1\end{pmatrix}
$$ 
Compute thee determinant of the linear transformation $T(X) =\frac{1}{2} (AX+XA)$.
Any hints would be appreciated.
 [/INST] Note that your transformation is realized by a $9\times 9 $ matrix, as $X$ has $9$ elements.
So 
$$
\left(
\begin{matrix}
x_{11} \\
x_{12} \\
x_{13} \\
x_{21} \\
x_{22} \\
x_{23} \\
x_{31} \\
x_{32} \\
x_{33}
\end{matrix}
\right) \quad\longmapsto\quad 
\left(
\begin{matrix}
x_{11} \\
\tfrac{3}{2}x_{12} \\
x_{13} \\
\tfrac{3}{2}x_{21} \\
2x_{22} \\
\tfrac{3}{2}x_{23} \\
x_{31} \\
\tfrac{3}{2}x_{32} \\
x_{33}
\end{matrix}
\right)
$$
and this is realized by the $9\times 9$ diagonal matrix
$$
T=\mathrm{diag}\,(1,\tfrac{3}{2},1,\tfrac{3}{2},2,\tfrac{3}{2},1,\tfrac{3}{2},1).
$$
Clearly $\det T=1\times\tfrac{3}{2}\times 1\times\tfrac{3}{2}\times2\times\tfrac{3}{2}\times1\times\tfrac{3}{2}\times1$
</s>