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I have a question which says this:
Define these actions on lines of matrices:
e1 = multiply the line I in scale $c≠0$
e2 = switch the line i with the line j
e3 = add $c *$ line $j$ to line i
prove you can perform e2 with the uses of e1 and e3.

Okay, what I did:
first, I wrote in math what each e means:
e1 = $Ri → c * Ri$
e2 = $Ri → Rj$
e3 = $Ri → Ri + c * Rj$
I couldnt think of anything I can do... If I could swap back line j with line i somehow ( it says from I to J, not opposite, so I cant use switch operation ). Any tip? no solution please, only a tip how can I start answering this question ( and if possible, if I wrote in math correctly the e1,e2,e3. its the first time I encounter a question of such.

  • so you mean I can do $Rj↔Ri$? –  Nov 09 '21 at 23:37
  • That is an elementary row operation. But the exercise asks how we can accomplish the "swap" using the other two kinds of elementary row operations. Note that replacing.one row with another is not an elementary row operation, and it would not generally preserve the matrix properties such as rank we want to preserve. – hardmath Nov 09 '21 at 23:45
  • Then I dont really understand what you mean. About that the 3e I have is the three row operation - yea I know. But the problem is, they said swap row i with row j, or they meant as in the operation row j ↔ row i? if yes, I think I might be able to solve it, is it true? the ↔? –  Nov 09 '21 at 23:48
  • I deleted some of my Comments for the sake of brevity. Think about the case of a matrix with just two rows. We want to swap those two rows, using only operations of type $e1$ and $e3$ without applying $e2$ for the purpose. You'll probably want to get pencil and paper and try a few things. – hardmath Nov 10 '21 at 14:56
  • You dont think I tried? But okay, I will delete my old comments also. but okay.. thanks anyway. –  Nov 10 '21 at 16:20
  • I think we are making progress, since you wanted a hint rather than a solution. The most important steps use $e3$; we only need to use $e1$ once, and that can be the final step. So focus on starting with $e3$. – hardmath Nov 10 '21 at 16:41
  • Wait a minute... I can use each operation a few times? I dont have to use it only once? I can do it lets say even 5 times? That I didnt know... Thanks, I will try it now. –  Nov 10 '21 at 16:49

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