# 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.

• Which book did you take the excerpt from? – PtF Aug 3 '18 at 14:22
• Have you looked at all the questions stackexchange lists in the margin as Related? This one might help: math.stackexchange.com/questions/1997523/… – Ethan Bolker Aug 3 '18 at 14:57
• Elementary Linear Algebra by Larson 7th edition @PtF – Jwan622 Aug 3 '18 at 18:46
• Thanks @Jwan622 – PtF Aug 3 '18 at 19:57
• Thoughts? Do you have a clarification? – Jwan622 Aug 4 '18 at 12:40

## 2 Answers

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

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.

• The question was why an invertible matrix can be row reduced to the identity matrix, not why row operations can be described by matrices. – md2perpe Aug 3 '18 at 14:45
• Yeah this answer while okay doesn't get at my question – Jwan622 Aug 4 '18 at 12:41
• @md2perpe See the last two paragraphs – mvw Aug 4 '18 at 17:47