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As I'm studying linear algebra I for my Computer Science and Physics degree, I was wondering to myself if anyone more experienced here could answer why the determinant is such an important part?

Where else would I encounter it ? and what does it help solve mostly?

devam_04
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ThomasReiner
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  • The utility/necessity of determinants is a contested topic! Determinants are useful for calculating inverses, volumes, computing eigenvalues, and.. determining invertibility. – Cameron Williams Jul 15 '21 at 02:40
  • So, it should be flaged as duplicate? – Jonatan Garcia Jul 15 '21 at 02:40
  • A could of quick uses: the determinant is a quick signal of linear dependence; the determinant is how one computes characteristic polynomials, which is how one typically computes eigenvalues, which are used all over; calculus is often about linear approximation, and in multivariable calculus the change of variables in integral formula distorts area by the determinant of a matrix associated by the change of variables. – davidlowryduda Jul 15 '21 at 02:41
  • Have a look at this. – ultralegend5385 Jul 15 '21 at 02:46
  • I have read all of the shared links. I appreciate the swift replies! However, I'm looking to understand not just the basic use of Det in Linear Systems, but rather, how it branches out into other fields like engineering, physics and computer science.

    So as far as Computer science goes, I can understand that it will be useful in solving complex linear equations efficiently and quickly. Can anyone elaborate on Eigenvalues ? I really enjoyed the 'Use of Determinants' post btw

     – ThomasReiner Jul 15 '21 at 02:49
  • https://www.google.com/books/edition/The_Theory_of_Determinants_in_the_Histor/DaZBAQAAMAAJ?hl=en&gbpv=1&printsec=frontcover – Will Jagy Jul 15 '21 at 03:14
  • You could write a whole book on the applications of determinants! It seems that you have a lot of ideas but they need to have more focus, so why not come back in a while and ask when you have a more concrete idea? – Toby Mak Jul 15 '21 at 03:15
  • Thank you Will! Book ordered, shame there is no Kindle version.. And thank you Toby, I def will! – ThomasReiner Jul 16 '21 at 02:18
  • People care a lot about the computation of the determinant, but that doesn't show how natural the determinant is, just how useful it is. The determinant is the unique, alternating, multilinear function on the columns of a matrix, and in fact uniquely encodes orientation. That is, the sign of the determinant of a matrix encodes the order in which the column vectors of that matrix are arranged in space. I consider this geometric property identifiable with the determinant as an example of why it is so special. – While I Am Jul 19 '21 at 17:04

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