We show that, for $\Omega$ a bounded convex domain of $\mathbb{R}^2$, any $2\times 2$ symmetric matrix $A(x)$ with $\det A(x)=1$ for a.e. $x\in\Omega$ satisfying the ellipticity bounds
$$\frac{|\xi|^2}{H}\le \langle A(x)\xi,\xi\rangle \le H|\xi|^2$$
for a.e. $x\in\Omega$ and for all $\xi\in\mathbb{R}^2$ can be approximated, in the sense of $G$-convergence, by a sequence of matrices of the type
$$\left(\begin{matrix}\gamma_j(x)& 0\\ 0&\frac{1}{\gamma_j(x)}\end{matrix}\right)$$
with $$H-\sqrt{H^2-1}\le \gamma_j(x)\le H+\sqrt{H^2-1}\,.$$