I want to know how we find or construct the matrix representations for Grassmann numbers.
For example, we can see from https://en.wikipedia.org/wiki/Grassmann_number:
Grassmann numbers can always be represented by matrices. Consider, for example, the Grassmann algebra generated by two Grassmann numbers 1 and 2. These Grassmann numbers can be represented by 4×4 matrices:
$$\theta_1 = \begin{bmatrix} 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0\\ 0 &
0 & 0 & 0\\ 0 & 0 & 1 & 0\\ \end{bmatrix}\qquad \theta_2 =
\begin{bmatrix} 0&0&0&0\\ 0&0&0&0\\ 1&0&0&0\\ 0&-1&0&0\\
\end{bmatrix}\qquad \theta_1\theta_2 = -\theta_2\theta_1 =
\begin{bmatrix} 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 1&0&0&0\\ \end{bmatrix}.$$
In general, a Grassmann algebra on $n$ generators can be represented by $2^n \times 2^n$ square matrices. Physically, these matrices can be thought of as raising operators acting on a Hilbert space of n identical fermions in the occupation number basis. Since the occupation number for each fermion is 0 or 1, there are $2^n$ possible basis states.
Does there exist a general method to construct these matrices if one has three or $n$ Grassmann numbers? How to connect the relation between the Grassmann numbers and the Ferminoic path integral on QFT? How to understand that "Physically, these matrices can be thought of as raising operators acting on a Hilbert space of $n$ identical fermions in the occupation number basis. Since the occupation number for each fermion is 0 or 1, there are $2^n$ possible basis states."?
