I'm working through a tutorial on matrices:

Pauls Online Notes : Linear Algebra - Finding Inverse Matrices
I'm confused about the following bit:

If $\displaystyle A$ is invertible then there are a set of elementry matrices $\displaystyle E_k...E_2E_1$ such that $\displaystyle E_k...E_2E_1A=I_n$. If we multiply both sides of this by $\displaystyle A^{-1}$:

$\displaystyle E_k...E_2E_1AA^{-1}=I_nA^{-1}\Rightarrow A^{-1}=E_k...E_2E_1I_n$

I'm confused by that statement. I don't see how to get to isolate the inverse to obtain $\displaystyle A^{-1}=E_k...E_2E_1I_n$. On the left side of the arrow, I have $\displaystyle E_k...E_2E_1AA^{-1}=E_k...E_2E_1I=I_nA^{-1}$, but I still don't see how this implies that $\displaystyle A^{-1}=E_k...E_2E_1I_n$