I got trace is $4$ for a $2\times2$ matrix. I know trace is a linear operation which transforms a matrix to a field. So the matrix representation of a trace should be like $1×n$, where $n$ is the dimension of that square matrix.
How I can represent number $4$ as a $1\times4$ matrix? Thanks in advance!
You can represent a $2\times2$ matrix $\begin{bmatrix}a&b\\c&d\end{bmatrix}$ by the $4\times1$ vector $\begin{bmatrix}a\\b\\c\\d\end{bmatrix}$, then the trace matrix is $\begin{bmatrix}1&0&0&1\end{bmatrix}$. We have$$\begin{bmatrix}1&0&0&1\end{bmatrix}\begin{bmatrix}a\\b\\c\\d\end{bmatrix}=a+d=\text{Tr}\begin{bmatrix}a&b\\c&d\end{bmatrix}$$
We can get to this matrix easily: the trace of $e_1=\begin{bmatrix}1\\0\\0\\0\end{bmatrix}$ is $1$, which is the first column of trace matrix. The trace of $e_2=\begin{bmatrix}0\\1\\0\\0\end{bmatrix}$ is $0$, which is the second column of trace matrix. Similarly the traces of $e_3,e_4$ are $0,1$ respectively.