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Geometric Interpretation of why row/column elementary operation doesn't change determinant?

I found this comment, which might potentially answer the question, but I'm not sure how to interpret it.

i first created a parallelogram

i scaled it

i further scaled it

where CD is taken to be $v_1$ and BC is taken to be $v_2$, scaling it by dragging point B and leaving the other points unchanged transforms $v_2$ to $v_2 + λv_1$, where $λ$ is a constant. the comment seems to imply that the area of the geometric figures in all my images will be equivalent, but it doesn't look the same to me... Especially since if we rotate the figure $180$ degrees, $AB$ can be taken as the base and it seems to have increased.

What am i doing wrongly? Any guidance is appreciated!

Bowei Tang
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algo
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  • ok, but the comment @Malkoun made in 2020, seems to provide an answer to this question, which I do not understand how to interpret... – algo Oct 02 '24 at 14:32
  • in addition, that original question was left unanswered, https://math.stackexchange.com/questions/3676802/geometric-interpretation-of-why-row-column-elementary-operation-doesnt-change-d, so i was wondering if anyone knows the answer? i can remove the reference to chatgpt, but that was one thing i did to try to find an answer...... – algo Oct 02 '24 at 14:33
  • The two "scaled" pictures 1 2 are no longer parallelograms! So they have nothing to do with determinants anymore. – Misha Lavrov Oct 02 '24 at 14:35
  • This question is similar to: What's an intuitive way to think about the determinant?. (Specifically, there is an answer with the visualizations I think you're looking for.) If you believe it’s different, please [edit] the question, make it clear how it’s different and/or how the answers on that question are not helpful for your problem. – Misha Lavrov Oct 02 '24 at 14:40
  • @MishaLavrov i had checked that when i was trying to understand other stuff, such as the part about scaling. but it doesn't say anything abuot row addition? – algo Oct 02 '24 at 14:43
  • @MishaLavrov - the original unanswered question they've mentioned in a comment? I'm just dealing with the ChatGPT part, but it looks like you've found an existing question that makes this a duplicate? That's a better resolution . – JonathanZ Oct 02 '24 at 14:45
  • @algo Check out the specific answer I recommend; it is specifically a visualization of row addition and area. – Misha Lavrov Oct 02 '24 at 14:50
  • @JonathanZ If you mean this one, then it's not closed, it's just unanswered. – Misha Lavrov Oct 02 '24 at 14:50
  • Ah, you're right, I misspoke. I think of old questions as more-or-less 'gone', as they are much less likely to receive answers than recent ones. I meant "put some attention back on", not "reopen ". – JonathanZ Oct 02 '24 at 14:55
  • @MishaLavrov thanks! im still processing the information but it looks promising – algo Oct 02 '24 at 15:00

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