Main Idea Behind Determinant & Its Purpose

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SUMMARY

The determinant is a mathematical concept that quantifies the signed volume of the N-dimensional parallelepiped formed by N vectors. It is primarily utilized for solving systems of equations and provides a straightforward method for obtaining a totally anti-symmetrized product. Although determinants are less frequently used in general linear systems due to the availability of more efficient methods, they remain valuable for specially-structured systems and theoretical applications.

PREREQUISITES
  • Understanding of linear algebra concepts
  • Familiarity with N-dimensional geometry
  • Knowledge of systems of equations
  • Basic proficiency in mathematical notation
NEXT STEPS
  • Research advanced linear algebra techniques
  • Explore alternative methods for solving systems of equations, such as Gaussian elimination
  • Study the geometric interpretation of determinants in higher dimensions
  • Investigate the applications of determinants in theoretical mathematics
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Mathematicians, students of linear algebra, and anyone interested in the theoretical and practical applications of determinants in solving equations.

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What is the main idea behind the determinant? What was the main purpose for which it was conceived?
 
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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.
 
BvU said:
Extremely useful for solving systems of equations. History here but I suppose you googled that too ?

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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