Question

If I reduce two matrix into reduced row echelon form and both of them becomes an identity matrix, does this mean they are both row equivalent?

Answer 1

Well, two matrices are row-equivalent if you can get from one to the other using elementary row operations (EROs). Now say two matrices A and B both reduce to the identity.

But EROs are reversible! So if you can get from B to the identity using EROs, then you can get from the identity to B also using EROs. So you can get from A, to the identity, to B, all using EROs. So the answer is yes.

Answer 2

In fact, if you can reduce two matrices to the *same* matrix (not necessarily the identity) using EROs, then the two original matrices are row equivalent.

Answer 3

thank you for your time you deserve a medal :)