What is the main idea behind the determinant? What was the main purpose for which it was conceived?
Extremely useful for solving systems of equations. History here but I suppose you googled that too ?
The determinant of of N vectors gives you the (signed) volume of the N-dimensional parallelepiped they span. Most of its uses come from either this property, or it's property as the simplest way of getting a totally anti-symmetrized product of stuff.
Actually, determinants rarely used anymore for solving general linear systems of equations, because there are so many more efficient and simpler methods available. However, for specially-structured systems, determinants can, indeed, be the best way of solving them. They are also very useful theoretically.
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