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The determinant of a matrix is how much the matrix expands or contracts space. More precisely: it's the volume of the parallelepiped spanned by the columns of the matrix. (Ditto the rows.) This, btw, is why det 0 => non-invertible: one or more spatial dimensions has collapsedhttps://twitter.com/ZachWeiner/status/1073671743846395905 …
That's one way. There are many other useful equivalent ways of thinking of the det: (1) As the product of the eigenvalues (for normal matrices); (2) as the product of the singular values (up to sign) for any matrix; (3) as the value of the wedge product of the columns c1^c2^ ...
(4) as a matrix function which (a) rescales when you rescale a column, (b) isn't changed by addition of one column to another, and (c) changes sign when columns are swapped. Fun to think through why all these are equivalent! Many other ways of thinking of the determinant, too.
Some considerable fraction of mathematics seems to be keeping multiple ways of thinking about an object in your head, and moving back and forth between them.
pshhhh you just copied my answer!https://twitter.com/GarrulousGeoff/status/1073672079801630720 …
Obviously, as you're the first person in history to notice that the determinant measures the expansion of space. (I don't know the origins of the determinant. But I presume this was it. Obviously whoever came up with the Jacobian - presumably Jacobi or a predecessor - knew it.)
Thank you for acknowledging my achievements 
How does this explain the signedness? Why does transposing one pair of columns flip the “sign” of the volume, while transposing two pairs doesn’t?
A negative sign means space has been reflected through a plane (in 3d, means a change in "handedness"). Doesn't change the absolute value of the volumes. This is best illustrated with a few clarifying examples, which I won't try to go through on twitter.
The part of the volume explanation that seems mysterious is: how does reflecting through an even number of planes differ from reflecting through an odd number of planes?
Reflecting through one plane changes a left hand into a right hand. Reflecting through two changes it back. And so on.
Changing left and right hand is a topological property. The determinant of a matrix A is negative, iff there is a continuous path A_t t \in [0,1] of _invertible_ matrices such that A_0 = A and A_1 = reflection in plane (a so called homotopy).
You know that one of the only two analytically solved Quantum Mechanics calculations deal with Hermetian matrices?
I'm not sure what you're referring to(?) - lots of quantum mechanics calculations have been analytically solved.
By hand.
I kinda thought I got determinants but realized how false that was when I wrote this explanatory piece on orthogonal matrices. Nothing more fun than having multiple "kinda but don't get it" areas suddenly click. Your explanation made even more clicks :Phttp://smerity.com/articles/2016/orthogonal_init.html …
I found the @3blue1brown explanation of determinants very clear - an animated version of @michael_nielsen's words above :)
@3blue1brown has probably the best graphical and intuitive explanation of what a determinant is!https://www.youtube.com/watch?v=Ip3X9LOh2dk …
@3blue1brown have this neatly visualized:https://www.youtube.com/watch?v=Ip3X9LOh2dk …
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