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I am not able to make sense of the determinant formula. i can see why it is true for 2d and 3d separately, but not, 3d as an extension of 2d, but the formula has that kind of a sense(of connection between determinant in a dimension and determinants in the previous dimension).

what i think the formula is saying: ignore one basis vector and then see the plane spanned by the rest of the basis vectors as a plane projected onto the X-Y plane or Y-Z plane etc. (depending on what column (or basis vector) was ignored, like if we are ignoring the first basis vector then the projection of the plane (spanned by 2nd, 3rd… and so on basis vectors) onto the Y-Z plane, if it’s 2nd, then X-Y plane.. and so on). And then find the determinant of this projection, and then if the column (or basis vector), that we ignored, had an odd column number, then multiple the determinant calculated by the x coordinate of the ignored basis vector, and if it’s even, multiply it by the negative of the x coordinate of the ignored basis vector. And then add them all up, and you get the determinant.

is this explanation correct?

How can someone even come up with this formula from scratch?

thank you

Edit : by 'determinant formula' I mean this formula: enter image description here

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    $\begingroup$ You mean the so-called Laplace expansion formula with respect to a row or column using cofactors ? $\endgroup$
    – Jean Marie
    Feb 24, 2022 at 16:39
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    $\begingroup$ What is the "determinant formula"? $\endgroup$
    – undefined
    Feb 24, 2022 at 18:13
  • $\begingroup$ @ScheffleraArboricola : The absolute value of a determinant is the area, resp. the volume of the parallelogram, resp. the prallel epiped, spanned by the vectors. $\endgroup$
    – Kurt G.
    Feb 24, 2022 at 18:24
  • $\begingroup$ I have added the determinant formula to the question $\endgroup$ Feb 25, 2022 at 10:54
  • $\begingroup$ I'd recommend following "3blue1brown" video series on Youtube about essence of linear algebra, you will definitely get some intuition there $\endgroup$
    – Dean
    Feb 25, 2022 at 12:26

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