Why can all invertible matrices be row reduced to the identity matrix? All my textbook has covered so far is this:

But then the textbook hits me with this:

which basically tells me that an invertible matrix can be row reduced to I (multiplying on the left by elementary matrices I just learned is equivalent to a series of row operations). Why is this? I don't really find this intuitive.
 A: Depending on which basics of linear algebra you've covered so far, it should be straightforward to check that:


*

*A matrix is invertible if and only if its row reduced echelon form is invertible

*A matrix in row reduced echelon form is invertible if and only if it is the identity matrix

A: 
which basically tells me that an invertible matrix can be row reduced
  to I (multiplying on the left by elementary matrices I just learned is
  equivalent to a series of row operations). Why is this? I don't really
  find this intuitive.

Is it unclear thar row operations can be expressed by multiplication with a elementary matrix? 
You should have a look of the elementary matrices and check for yourself, that multiplication with them gives the same result as a row operation would.
$$
E A = A'
$$
So if $E$ is the elementary matrix for swapping certain two rows then $A'$ should be the same as $A$ having those two rows swapped.
Or is it unclear why $A$ can be expressed as a product of the inverses  of elementary matrices (acting on $I$)? Or equivalent: $A^{-1}$ can be expressed as product of elementary matrices.
This is because the Gauss elimination procedure is basically acting like a multiplication with an inverse. And this can be expressed as product of elementary matrices. 
