Three proofs with row-equivalent matrices

Question

I am struggling with these proofs, since I haven't done any proofs with row-equivalent matrices before, so I cannot get the idea behind it really. I was hoping you could help me out to get on the right track!

1. Prove that two m x n matrices A and B are row-equivalent if and only if there exists an invertible m x m matrix U with A = UB.

2. Prove that A is row-equivalent with itself.

3. Prove that if A and B are row-equivalent and B and C are row-equivalent, then A and C are also row-equivalent.

I thought that you might need elementary matrices to solve this. Am I right?

Also, regarding the second proof, this seems very obvious. One can create a matrix out of the same matrix. But how can you officially prove this?

Thank you!

Answer 1

Hints.

1. Any invertible matrix is a product of elementary matrices
2. Multiplying a matrix (on the left) by an elementary matrix is the same as applying a suitable row operation.

How do you prove hint 1? Let $F$ be invertible; then its reduced row echelon form is the identity. Now apply hint 2 (that you should know about).