  In this paper, we study the action of special $n\times n $ linear (resp.
symplectic) matrices which are homotopic to identity on the right invertible
$n\times m$ matrices. We also prove that the commutator subgroup of
$\rm{O}_{2n}(R[X])$ is two stably elementary orthogonal for a local ring $R$
with $\frac{1}{2}\in R$ and $n\geq 3.$
