A transformation can be undone exactly when it preserves enough information to distinguish every input.
Multiplying the displayed candidate by A gives the identity when ad − bc is nonzero. A zero determinant collapses directions, losing information and preventing an inverse.
Dividing by a tiny determinant can amplify numerical error even when an inverse exists mathematically.
The recovered vector is A⁻¹(Av), computed from the selected vector and matrix.