TITLE: Determinant of a Linear Transformation
QUESTION [0 upvotes]: I need to find the determinant of the linear transformation 
from the space V of symmetric 2*2 matrices to V.
I've tried to use T*M = DM + MD and then
T = (DM+MD)*(M^-1)   
To find that det (T-D) = det (D) since MDM^(-1) is a similar matrix.
Does anyone have an answer, or any clue as to how I should tackle this question?

REPLY [1 votes]: Hint: choose a basis for the vector space of $2\times 2$ matrices, e.g. $$e_1=\pmatrix{1&0\\0&0}, e_2=\pmatrix{0&1\\0&0},e_3=\pmatrix{0&0\\1&0},e_4=\pmatrix{0&0\\0&1}$$
Then express each $T(e_i)$ as a linear combination of the basis matrices; for example, for $e_1$ we would get
\begin{align*}
T(e_1)&=\pmatrix{4&1\\1&-1}\pmatrix{1&0\\0&0} +\pmatrix{1&0\\0&0}\pmatrix{4&1\\1&-1}\\
&=\pmatrix{4&0\\1&0}+\pmatrix{4&1\\0&0}\\
&=\pmatrix{8&1\\1&0}\\
&=8e_1+1e_2+1e_3+0e_4
\end{align*}
Do this for $T(e_2), T(e_3)$, and $T(e_4)$ as well, and then you can write down the matrix of $T$ with respect to $e_1,e_2,e_3$, and $e_4$, which will be a $4 \times 4$ matrix, and its determinant will be what you are looking for.
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