Assume that $ A $ is a real $ n\times n $ matrix whose rows constitute an orthonormal basis of $ \mathbb R^n $.

**Informal statement of question:** Assume we want to approximate $ A $ by a rational matrix, such that each entry can be written efficiently (that is, has a small binary encoding), but we require also the inverse of the approximate matrix to have small representation. Is this possible?

**Formal statement of question:** Let $ p(n) $ be some polynomial in $ n $. For a real number $ r $, we say that $ a/b $ is a *polynomial approximation of $ r $*, if $ a/b$ is a rational number (that is, $ a,b $ are integers) and both $ a $ and $ b $ are of size at most $p(n) $ (e.g., their binary representation is of logarithmic size in $ n $), such that $ |r-a/b|\le 1/p(n) $.

*Question*: Does there exist a rational matrix $ B$, such that $ B $ polynomially approximates $ A $ (that is, the entry $ B_{ij} $ in $ B $, is a polynomial approximation of the entry $ A_{ij} $ in $ A $, for all $ 1\le i,j\le n $), and such that $ B^{-1} $ is a rational matrix whose entries are all polynomially-bounded (that is, for any $ 1\le i,j\le n $, $ B^{-1}_{ij}=a/b$, where $ a,b $ are integers of size at most $ p(n) $) ?