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\begin{aligned}|A|={\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}&=a\,{\begin{vmatrix}e&f\\h&i\end{vmatrix}}-b\,{\begin{vmatrix}d&f\\g&i\end{vmatrix}}+c\,{\begin{vmatrix}d&e\\g&h\end{vmatrix}}\\[3pt]&=aei+bfg+cdh-ceg-bdi-afh.\end{aligned}

Why do we choose to solve a 3*3 determinant like this? By "certain way" or "like this" I mean to ask-

  1. Why is it "a" times this derterminant {\begin{vmatrix}e&f\\h&i\end{vmatrix}
  2. Why do we follow a certain sign scheme
  3. What do we actually mean by solving a 3*3 determinant. Does the operation of solving a determinant has a meaning in itself?

PS: I am a high school student. I may have made a lot of errors while framing the question or the question may be unclear. Comments and corrections are most welcome

Ipsy-Doopsy
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  • The 3x3 determinant of a matrix represents the amount the volume of a region changes after being transformed by that matrix. The sign tells us whether the transformation preserves the orientation of the region. To prove the formula for the 3x3 determinant, try looking up the $\textit{triple product}$. – mpnm Mar 16 '22 at 16:57
  • Triple product looks beyond my scope of understanding at present as I have not started studying vector mathematics. But thanks, will look into it in the near future. – Ipsy-Doopsy Mar 16 '22 at 17:02
  • See https://en.m.wikipedia.org/wiki/Laplace_expansion – Shahab Mar 16 '22 at 17:06
  • @Ipsy-Doopsy Don't worry, it looks scary but all it does is calculate the area of a parallelpiped, a kind of 3 dimensional parallelogram. – mpnm Mar 16 '22 at 17:06
  • The punchline is that the problems we wish to solve come first and the definitions and approaches are created in such a way to be useful for solving those problems. Determinants are used for various things such as volumes, or even more generally deciding whether a square matrix is invertible or not (*if it is zero it is not invertible*). Multiple formulas exist which can be proven are equivalent to one another. The one you allude to is that of [Laplacian Cofactor Expansion](https://en.wikipedia.org/wiki/Laplace_expansion). – JMoravitz Mar 16 '22 at 17:08
  • Also popular is the [Leibniz formula for determinants](https://en.wikipedia.org/wiki/Leibniz_formula_for_determinants). Which is more useful depends on context and what it is you wish to show, but both will as mentioned already give the same final answer. – JMoravitz Mar 16 '22 at 17:10
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    About 3: "Solving a determinant" is simply not correct language. – Qi Zhu Mar 16 '22 at 17:21
  • “Solving” a determinant isn't the correct terminology. You can solve problems or equations, but for a determinant you would use “compute” or “calculate” or “evaluate” or something like that. – Hans Lundmark Mar 16 '22 at 17:40

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