Constructing private matrix for Algebraic Eraser
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The matrix $M_0$ is chosen from the set of $n\times n$ invertible matrices over $GF(q)$ (a group under multiplicatin) by some "proprietary" method.

A private key polynomial 
$$\sum_{i=0}^{N-1} \alpha_i x^i,$$
over $GF(q)$ is specified and $M_A$ is this polynomial evaluated at the matrix $M_0:$

$$M_A = \sum_{i=0}^{N-1} \alpha_i  M_0^i$$