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I am not able to understand why we expand the determinants the way we do, I have seen several applications of determinants and its usefulness but I don't get how it's able to solve so many geometrical problems so efficiently.

I'm currently studying in high school so other question-answers are not helpful for me

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  • $\begingroup$ Almost duplicate: math.stackexchange.com/questions/668/… $\endgroup$
    – Kyky
    Apr 17, 2021 at 10:44
  • $\begingroup$ Many people would argue that determinants ought to be defined in terms of geometry, rather than through some formula. See here. I would also highly recommend 3blue1brown's video series on linear algebra. $\endgroup$
    – user883638
    Apr 17, 2021 at 10:46
  • $\begingroup$ Also see here: m.youtube.com/watch?v=IxNb1WG_Ido $\endgroup$
    – Kyky
    Apr 17, 2021 at 10:51
  • $\begingroup$ The initial introduction to determinants in high school is deeply mysterious. Especially the fact that it can be evaluated along any row or column in a specified manner and result remains same. This was one of the biggest mysteries for me when I was in high school. You need to have patience and wait for a proper course in linear algebra. Of if you are adventurous enough do try to get hold of Hoffman & Kunze Linear Algebra. $\endgroup$
    – Paramanand Singh
    Apr 17, 2021 at 13:05

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Determinants and matrices are a topic of linear algebra, so their significance lies in linear equations.

Take the system of linear equations: $$a_1x+b_1y=c_1$$ $$a_2x+b_2y=c_2$$ In matrix form, this is the same as $$\underbrace{\begin{bmatrix} a_1 & b_1 \\ a_2 & b_2 \end{bmatrix}}_{A\ \text{(let)}} \begin{bmatrix} x\\y\end{bmatrix}=\begin{bmatrix}c_1\\c_2\end{bmatrix}$$ Now, whether or not they have a unique solution is determined by the number $a_1b_2-a_2b_1$. (You're in high school, so I assume you know how to solve linear equations; if you don't know, then you can learn! Comment, I will add some more info on that) So, we define this number to be the determinant of the matrix $A$, denoted by $\det A$ or $|A|$.

We can then generalise and utilise this method to solve any system of linear equations. You'll learn this later on in the same topic (at least I think).

Hope this helps. Ask anything if not clear :)

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