Linear Algebra 14TBD: The Transpose Property of the Determinant

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[@MathTheBeautiful]

Go to LEM.MA/LA for videos, exercises, and to ask us questions directly.

[@TheLeontheking]

I can only recommend watching to the end, even if you think you understand before - the close observation of the sign is very insightful! Everyone who studied permutations in the context of abstract algebra will enjoy the simple rearrangements presented here.

[@cardinalblues7121]

This is amazing, thank you so much for revealing the beauty of the mathematics. why not upload videos for Calc I. I am looking forward to your precious videos

[@MFMegaZeroX7]

I've really been enjoying this series up until now, but I guess I don't really have the proper permutation background here. The only thing I ever really learned about permutations was the formula used for things like 5 choose 3. Where would you recommend I go to brush up on this stuff?

[@dannuttle9005]

I was about to ask the same question, for the same reasons. I understand permutations, from probability, but I don't understand where parity comes from.

[@dannuttle9005]

This is not a complete explanation of what we see here but it helps explain whether a permutation is odd or even, plus it is based on a Futurama bit: kzbin.info/www/bejne/rWHQqZelatqllaM

[@PyMoondra]

Yeah I don't have a background as well, I just rolled with it as he did touch up the theorems as he explained the lesson.

[@TheLeontheking]

For other people stumbling upon this: I can recommend "A Book of abstract Algebra" by "C. Pinter" which has an excellent chapter about permutations. Depending on how curious you are it might be overkill though - there are probably other resources that let you learn about them. For starters, just think of them as a rearrangement of a finite number of elements. So if you start with, say: (1,2,3,4,5), then (5,4,3,2,1) would be one of the permutations. The book mentioned above of course puts this notion into the concept of sets and groups, which might or might not be helpful to you :)

[@atnn4698]

I just can not see it :( I try to see other proofs for this property, but they all use other properties (|AB|=|A||B|, Affect of row operations on determinants etc'...). There is NOTHING in this algebraic formula which somehow shows me that |A|=|At| FOR THE GENERAL CASE. Easy to see for 3X3, O.K, but we can not do it further, 4X4 and so on . . .I guess IM HOPELESS in the case of determinants

[@MathTheBeautiful]

Hi Atn, I think this follows most easily from the direct algebraic definition of the determinant. It's pretty easy to see that the n! terms are the same (in absolute value). To see that all the signs are the same, you'll have to think a little bit about how the two permutations for the corresponding terms are related.

[@dacianbonta2840]

the segment crossing argument nicely worked out Aitken's Determinants and Matrices that flopping the diagram upside down maintains the number of crossings can be seen as correct by a middle schooler

[@MathTheBeautiful]

Thanks for sharing!

[@dereksmetzer2039]

if you base the permutations with respect to, say, 23451 for the 5x5 case will the determinant still be equivalent to one where the base permutation was 12345, as shown in the video? i understand the signs will be different, but my intuition wants to say that the cumulative effect from the transpositions will either produce a determinant with the same value or an inversed value between cases.

[@enashameed471]

Thanks جزاك الله خيرا ❤

[@marxman1010]

Aij is permutate j to i, Aji is permutate i to j, so they are inverse permutation.