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**Defunkt** Test 2, Q1:

Test 2, Q4:

Your proof is incorrect. You are saying, essentially, that if A,B are row equivalent then $\displaystyle A=E_1E_2...E_k$ where $\displaystyle E_1E_2...E_k=B$ but then $\displaystyle A=E_1E_2...E_k=B \Rightarrow A=B$ and clearly not any two row equivalent matrices are equal. The correct formulation is that $\displaystyle A=E_1E_2...E_kB$ with $\displaystyle E_i$ elementary matrices.

You should also note that this proposition is not true: Take $\displaystyle A=I_2$ and do the row operation $\displaystyle R_1 \leftrightarrow R_2$ to get B, then: $\displaystyle det(A) = 1, det(B) = -1 \neq det(A)$