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In my incursion in the world of linear algebra there are a few concepts that I just can't wrap my head around, one of them being the definition of a determinant via Laplace expansion. Is anyone able to offer the intuitive idea of it ?

Math apprentice
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    You might find this post and this post to be helpful, even though they're not specifically geared towards the Laplace expansion formula – Ben Grossmann Aug 02 '22 at 14:53
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    It's not intuitive at all. – Qiaochu Yuan Aug 02 '22 at 14:53
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    What made the determinant finally click for me was to understand the definition as a multilinear alternating form, which can be motivated, and then see that they are uniquely given - up to a constant - by the Leibniz permutation formula for the determinant. From this formula, and this definition, you can derive many of the determinant properties, including the Laplace cofactor expansion – FShrike Aug 02 '22 at 14:56
  • @BenGrossmann thank you – Math apprentice Aug 02 '22 at 14:56
  • @QiaochuYuan well I know it's pretty hard to explain in a few words, but he must've had a reason, and I can't stand to move on in math without understanding the motivation for a concept. – Math apprentice Aug 02 '22 at 14:59
  • A matrix determinant is the signed volume of a parallelipiped made by its column vectors (and also line vectors). This property is sufficiently important (in my opinion) to justify the formula, and gives an intuitive explanation as to why the matrix rank is linked to the value of its determinant. – Jean-Armand Moroni Aug 02 '22 at 15:15
  • This topic has been beaten to death on this site, but you might find my video useful as it derives the Laplace expansion from a simple geometric result. – blargoner Aug 02 '22 at 16:11
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    I'd just like to comment that I'm not a history of maths expert, but I highly doubt that Laplace defined the determinant with the expansion formula. Rather, I guess he proved (or popularized) the formula, and maybe now there are textbooks which use it as a definition (I hope not, that would be a terrible definition). Now of course we can backtrack and find elegant and interesting justifications for the formula, but in the end it was established because it's a sometimes useful and practical way to compute determinants (also it's how you prove the formula for the inverse of a matrix). – Captain Lama Aug 02 '22 at 17:34
  • Here is a nice discussion of the History of the Determinant: https://mathshistory.st-andrews.ac.uk/HistTopics/Matrices_and_determinants/ – Disintegrating By Parts Aug 03 '22 at 06:21

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