TITLE: Need help with writing proofs for matrices that are row equivalent
QUESTION [0 upvotes]: I have two proofs I do not know how to start:
Q1: Prove that if A is row-equivalent to B and B is row-equivalent to C, then A is row-equivalent to C.
Q2: Let A be a nonsingular matrix. Prove that if B is row-equivalent to A, then B is also nonsingular. 
Q1 comes with some advice: 
"Getting started: To prove that A is row-equivalent to C, you have to find elementary matrices $$E_1E_2… E_k$$ such that A = $$E_k… E_2E_1C$$.
I. Begin by observing that A is row-equivalent to B and B is row equivalent to C.
II. This means that there exist elementary matrices $$F_1F_2… F_n$$ and $$G_1 G_2… G_m$$ such that A = $$F_nF_{n-1}… F_1B$$ and B = $$G_mG_{m-1}… G_1C$$.
III. Combine the matrix equations from step II."
I am unsure what row equivalence means and have no idea how A having an inverse makes B have an inverse if it is row equivalent; I've always froze up at proof questions for some reason so please do not be afraid to be verbose; I'd like to be able to explain and understand this stuff someday such that I will be able to write proofs properly.

REPLY [1 votes]: $$A=F_n...F_1B$$
$$B=G_n...G_1C$$
$$A=F_n...F_1G_n...G_1C$$
This shows that matrix A is equal to matrix C as B is equal to C through elementary row operations.
As for proving that if A is nonsingular, B is also nonsingular if row-equivalent to A...
B ~ A
$$T_n...T_1B = A$$
$$T^{-1}(T_n...T_1B) = T^{-1}(A)$$
$$B = T_n^{-1}...T_1^{-1}A=E_1...E_nA$$
Should show that B is also row-equivalent.
{"set_name": "stack_exchange", "score": 0, "question_id": 2921141}