What actually is a determinant? How it have such properties and what does it tell about?

Question

I have completed by Matrices classes and the only question that is roaming in my mind is what is this determinant actually. Like they simply told the formula and things that are to be calculated and how to calculate them.

I wanna know what these determinants actually tell about a matrix and how do they relate to them? Like we have a matrix and its determinant, how is this determinant related to it? And why do we calculate the determinant in the way we calculate it?

Like already matrix multiplication is a bit different then I expected it to be. One more thing I wanna know is why the determinant has such properties and how does it exhibit them? And how can these things be related to each other? Thanks!

Welcome to Mathematics Stack Exchange. A matrix describes a linear transformation, and its determinant is its volume scaling factor. The sign of the determinant tells whether the transformation preserves or reverses the orientation of n-space. — J. W. Tanner, Dec 10 '19 at 06:07
@Arthur I answer in the comments when I don't feel like providing a fully fleshed out answer. — Rushabh Mehta, Dec 10 '19 at 06:32
@DonThousand IMO, a brief answer is still an _answer_, more than it is a request for clarification or something like that. I think it is better for the site hygiene to have them in the answer section. — Arthur, Dec 10 '19 at 06:38
site hygiene might be the funniest terminology I've heard all day. — Rushabh Mehta, Dec 10 '19 at 06:39
Consider this: https://math.stackexchange.com/questions/3328630/demystifying-the-determinant — D.R., Dec 10 '19 at 06:43

score 1 · Accepted Answer · answered Dec 10 '19 at 06:26

An $n\times n$ matrix with real entries describes a linear transformation of $\mathbb R^n$. It maps the unit $n$-cube to an $n$-dimensional parallelotope (a parallelogram if $n=2$; a parallelepiped if $n=3$). The magnitude of the determinant is the $n$-dimensional volume of this parallelotope; thus, it describes the $n$-dimensional "volume scaling factor" of the linear transformation. The sign of the determinant indicates whether the transformation preserves or reverses orientation. If the determinant is zero, then the image parallelotope has volume zero and dimension less than $n$, so the linear transformation is not invertible.

What actually is a determinant? How it have such properties and what does it tell about?

1 Answers1