Determinants of matrices greater than 3x3

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Discussion Overview

The discussion centers on methods for calculating the determinant of matrices larger than 3x3, exploring both theoretical and practical approaches. It includes considerations of efficiency and manual computation techniques.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant expresses curiosity about finding the determinant of 4x4 or larger matrices.
  • Another participant references the Laplace expansion as a theoretical method but critiques its inefficiency for larger matrices, noting it requires n! operations.
  • A participant proposes using row operations to simplify the matrix to upper triangular form, stating that this method is significantly more efficient, requiring about n³ operations.
  • Some participants share their personal experiences with calculating determinants manually for n ≥ 3, indicating a preference for the row operation method.
  • There is a request for an example of how to perform row operations to find the determinant.
  • One participant mentions that they are self-teaching multidimensional mathematics and have only learned the row operation method so far.

Areas of Agreement / Disagreement

Participants generally agree on the inefficiency of the Laplace expansion for larger matrices and support the use of row operations as a more efficient method. However, there is no consensus on the best approach for arbitrary n determinants, as some participants express differing levels of familiarity and preference for methods.

Contextual Notes

Limitations include the lack of detailed examples or step-by-step explanations for the row operation method, as well as the absence of a comprehensive discussion on other potential methods for calculating determinants of larger matrices.

Who May Find This Useful

This discussion may be useful for students or individuals interested in matrix theory, particularly those looking for efficient methods to compute determinants of larger matrices.

smize
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I am wondering how one would find a the determinant of a 4x4 or greater. This isn't an urgent question, just a curiosity.
 
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voko said:
http://en.wikipedia.org/wiki/Laplace_expansion

That is "correct", and an interesting theoretical result, but it's a hopelessly inefficient way to calculate the determinant of a general matrix, because it takes of the order of n! operations for an n x n matrix.

A much more efficient way is to do row operations on the matrix which don't change the value of the determinant (or only multiply it by -1), but systematically change the matrix so that all the entries below the diagonal are zero. The determinant is then just the product of the diagonal terms. In the worst case, that takes about n3 operations. For a 10 x 10 matrix, n3 = 1,000 and n! = about 3.6 million, so one way is about 3600 times faster than the other!
 
AlephZero said:
A much more efficient way is to do row operations on the matrix which don't change the value of the determinant (or only multiply it by -1), but systematically change the matrix so that all the entries below the diagonal are zero. The determinant is then just the product of the diagonal terms. In the worst case, that takes about n3 operations. For a 10 x 10 matrix, n3 = 1,000 and n! = about 3.6 million, so one way is about 3600 times faster than the other!

By chance, could you give an example of how to do row operations to find the determinant?
 
AlephZero said:
That is "correct", and an interesting theoretical result, but it's a hopelessly inefficient way to calculate the determinant of a general matrix, because it takes of the order of n! operations for an n x n matrix.

Whenever I get to compute a det with n >= 3 MANUALLY, I use this method. Possibly because I remember it by heart. Doing arbitrary n dets most efficiently is a distinctly different matter.
 
voko said:
Whenever I get to compute a det with n >= 3 MANUALLY, I use this method. Possibly because I remember it by heart. Doing arbitrary n dets most efficiently is a distinctly different matter.

At the moment, it is the ONLY method I have learned. (I am self-teaching myself Multidimensional Mathematics until classes start in 3 weeks).
 

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