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What's an intuitive way to think about the determinant?

I just learned the basics of matrices. Then I came across the magical formula $$\det(AB)=\det(A)\det(B)$$

I understand the proof, but I don't really understand why it is true. It seems that we are too lucky. The first step is to prove that $\det(EA)=\det(E)\det(A)$ for elementary matrix $E$. How come we are so lucky that this is true for all three types of elementary matrices?

How did they come up with determinants in the first place? I understand that determinant was originally used to determine the number of solutions of linear equations. But how did they come up with such intricate formula?

Please enlighten me up with intuitive explanations. I'm tired with all the formal proofs.