I had linear algebra a long time ago, and I feel I can do all the computations. However, I still don't know why, for instance, a determinant tells me a system of equations is solvable. I don't know why the determinant gives me the volume of a parallelepipeds.

Most textbooks on linear algebra seem to just skim over this part. The books may tell you why a result is true for the 2x2 matrix, but not the general case.

What book should I read to learn the logic behind concepts like the determinant?

Martin Sleziak
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  • You should try googling it first. [What I found](https://books.google.co.in/books?id=O8KCBgAAQBAJ&pg=PA303&dq=determinant+volume+of+parallelepiped+linear+algebra&hl=en&sa=X&ved=0ahUKEwiCzJqy4uLJAhVCC44KHbgBCWMQ6AEINjAF#v=onepage&q=determinant%20volume%20of%20parallelepiped%20linear%20algebra&f=false). You should change the title of the question as well. It is very misleading. –  Dec 17 '15 at 11:17
  • [Wy determinant is volume of parallelepiped in any dimensions](http://math.stackexchange.com/questions/427528/why-determinant-is-volume-of-parallelepiped-in-any-dimensions) and [Determinants and volume of parallelotopes](http://math.stackexchange.com/questions/750/determinants-and-volume-of-parallelotopes). (You can probably find more posts about this.) – Martin Sleziak Dec 17 '15 at 12:00
  • My question is more general than that. The derminant is an example. Maybe I should edit the question to make that more clear. – Avatrin Dec 17 '15 at 12:03
  • Maybe it is worth checking some other posts tagged [book-recommendation+linear-algebra](http://math.stackexchange.com/questions/tagged/book-recommendation+linear-algebra) or [reference-request+linear-algebra](http://math.stackexchange.com/questions/tagged/reference-request+linear-algebra). – Martin Sleziak Dec 17 '15 at 13:48
  • You could try *Introduction to Linear Algebra* by Lang. It carries out the proofs in full in the $2 \times 2$ and $3 \times 3$ cases. For arbitrary $n$, the full proofs are given in his *Linear Algebra*, but the details might be a bit technical for some people. – David Dec 17 '15 at 14:04

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