[Math] Why can all invertible matrices be row reduced to the identity matrix

Question

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.

Answer 1

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