Let $K$ be a field. For $1\leq i,j\leq n$ such that $i\ne j$ and $\lambda\in K$, let $B_{ij}(\lambda):=I_n+\lambda E_{ij}$, where $E_{ij}$ is the matrix whose $(i,j)$-th entry is $1$ and is $0$ otherwise. I want to show that all matrices of the form $B_{ij}(\lambda)$ are generated by $B_{12}(\lambda)$ and permutation matrices.
My idea is to show that by left and right multiplication by transposition matrices it is possible to turn $B_{12}(\lambda)$ into $B_{ij}(\lambda)$. Multiplying $B_{12}(\lambda)$ by $M(\tau_{2i})$ on the left and $M(\tau_{1j})$ on the right gives a matrix whose $(i,j)$-th entry is $\lambda$. Then by multiplying this matrix by transposition matrices not moving the $i$-th row or the $j$-th column we can arrive at $B_{ij}(\lambda)$.
The problem is that I am not sure how to make this proof rigorous. Should I be using induction? Any suggestions to get me started?
Edit:
Note that $\tau_{ij}$ denotes the transposition with support $\{i,j\}$ and $M(\tau_{ij})$ denotes its matrix.