If $A$ is a nonsingular matrix, then $A=E_n...E_2.E_1$, where $E_n$ are elementary matrices

Question

How do I prove that

If $A$ is a nonsingular matrix, then there exists elementary matrices $E_1,E_2,E_3....E_n$ such that, $$ A=E_n...E_3.E_2.E_1.I=E_n...E_3.E_2.E_1 $$

My Understanding:

I feel this got to be true from the row operations that we use to find $A^{-1}$ from the equation $A=IA$, but how do I prove it mathematically ?

Answer 1

Hint:

Remember that every elementary operation on the rows of $\;A\;$ is a product $\;EA\;$ ,where $\;E\;$ is an elementary matrix. Observe $\;E\;$ multiplies from the left, otherwise that'd be an elementary operation on the columns of $\;A\;$ .

Try to take it from here.