# Finding a matrix representation for two Grassmann numbers.

This question is more general in the sense that I want to know how one finds a particular (say matrix) representation for any object. For the case of Grassmann numbers we have from Wikipedia the following representation:

Grassmann numbers can always be represented by matrices. Consider, for example, the Grassmann algebra generated by two Grassmann numbers $\theta_1$ and $\theta_2$. These Grassmann numbers can be represented by 4×4 matrices:

$$\theta_1 = \begin{bmatrix} 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0\\ \end{bmatrix}\qquad \theta_2 = \begin{bmatrix} 0&0&0&0\\ 0&0&0&0\\ 1&0&0&0\\ 0&-1&0&0\\ \end{bmatrix}\qquad \theta_1\theta_2 = -\theta_2\theta_1 = \begin{bmatrix} 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 1&0&0&0\\ \end{bmatrix}.$$

How do one find these matrices? Do you guess them or is there a procedure? What about finding differenct matrix-representations for Dirac $\gamma$-matrices? How do you find them?