Why is the product of elementary matrices necessarily invertible? Why is the product of elementary matrices necessarily invertible? I understand that each elementary matrix is invertible, but why is their product also invertible? Is it the indirect result of this theorem?


That $A$ is invertible means precisely that there is another matrix $A^{-1}$ such that
$$AA^{-1}=I=A^{-1}A.$$
It is easy to show that the inverse, if it exists, must be unique.
Now suppose $A,B$ to be invertible, and denote $C:=AB$. $C$ will be invertible if we can find $C^{-1}$ such that $CC^{-1}=I=C^{-1}C$. Observe that $$C(B^{-1}A^{-1})=(AB)(B^{-1}A^{-1})=A(BB^{-1})A^{-1}=AIA^{-1}=AA^{-1}=I.$$
Similarly we get $(B^{-1}A^{-1})C=I$. Therefore, since inverses are unique, by definition we can conclude that $C^{-1}=B^{-1}A^{-1}$.
