Let $G$=$\text{GL}_n(K) $ ($K$ is a field), $B$ is the subset of upper triangular matrices and $N$ is the subset of monomial matrices (or generalised permutation matrices).
I am trying to show that given $x \in G$, there are $b_1, b_2 \in B $ such that $b_1xb_2 \in N$ which would imply the result in after.
I know that by performing elementary row operations and column operations we can bring $x$ to smith normal form which in this case will give the identity matrix - this corresponds to multiplying $x$ on the left and right by elementary matrices and so certainly there are $e_1, e_2 \in G$ such that $e_1 x e_2 \in N $. However, the elementary matrices are only triangular and not necessarily upper triangular. Is there a way of only using upper triangular elementary matrices instead to give the result?