We are given a continuous function $\xi : \mathbb{R} \rightarrow \mathbb{R}$ satisfying condition (V.1) (with $d \geqslant 2$), where $$\forall A \in \mathcal{M}_d(\mathbb{R}), \quad A \text{ invertible} \Rightarrow f_\xi(A) = \left(\xi(A_{i,j})\right)_{1\leqslant i,j\leqslant d} \text{ invertible} \tag{V.1}$$ Show $$\forall (a, b, c, d) \in \mathbb{R}^4, \quad ad \neq bc \Rightarrow \xi(a)\xi(d) \neq \xi(b)\xi(c)$$ One may consider the matrix $\begin{pmatrix} a & b & 0 & \cdots & 0 \\ c & d & 0 & \cdots & 0 \\ c & d & & & \\ \vdots & \vdots & & I_{d-2} & \\ c & d & & & \end{pmatrix}$.
We are given a continuous function $\xi : \mathbb{R} \rightarrow \mathbb{R}$ satisfying condition (V.1) (with $d \geqslant 2$), where
$$\forall A \in \mathcal{M}_d(\mathbb{R}), \quad A \text{ invertible} \Rightarrow f_\xi(A) = \left(\xi(A_{i,j})\right)_{1\leqslant i,j\leqslant d} \text{ invertible} \tag{V.1}$$
Show
$$\forall (a, b, c, d) \in \mathbb{R}^4, \quad ad \neq bc \Rightarrow \xi(a)\xi(d) \neq \xi(b)\xi(c)$$
One may consider the matrix $\begin{pmatrix} a & b & 0 & \cdots & 0 \\ c & d & 0 & \cdots & 0 \\ c & d & & & \\ \vdots & \vdots & & I_{d-2} & \\ c & d & & & \end{pmatrix}$.