Main Idea Behind Determinant & Its Purpose

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    Determinant Idea
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Discussion Overview

The discussion centers on the main idea behind the determinant in mathematics, exploring its purpose and applications, particularly in relation to solving systems of equations and its geometric interpretation.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant asks about the main idea and purpose of the determinant.
  • Another participant claims that determinants are extremely useful for solving systems of equations, suggesting a historical context.
  • A different participant explains that the determinant of N vectors represents the signed volume of the N-dimensional parallelepiped they span, highlighting its geometric significance.
  • One participant notes that while determinants are useful for solving specially-structured systems, they are rarely used for general linear systems due to the availability of more efficient methods.
  • This same participant also mentions that determinants have theoretical usefulness.

Areas of Agreement / Disagreement

Participants express differing views on the current utility of determinants for solving systems of equations, with some emphasizing their historical importance and others noting their declining use in favor of more efficient methods. The discussion remains unresolved regarding the overall relevance of determinants in modern applications.

Contextual Notes

Some assumptions about the context in which determinants are used may be missing, and there is a lack of consensus on their current applicability in solving linear systems.

vktsn0303
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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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