SCHWEITZER 1961 Problem 5

Determine the functions $G$ defined on the set of all non-zero real numbers the values of which are regular matrices of order $2$ , and the functions $f$ mapping the two-dimensional real vector space $E_2$ into itself, such that for any vector $y \in E_2$ and for any regular matrix $X$ of order $2$ , $f(X_y)= G(det X)Xf(y)$ ( $det X$ denotes the determinant of $X$ ).(A. 5)