Understanding and Calculating Matrix Determinants - Step by Step Guide

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    Determinant Matrix
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SUMMARY

The determinant of the given matrix, defined as a square matrix with elements of the form 1-n on the diagonal and 1 elsewhere, is conclusively 0 for any order n. This is demonstrated through row reduction techniques, specifically by transforming the matrix into a triangular form, which reveals a row of zeros. The process involves adding all rows to the first row, leading to a determinant of zero for matrices of size n=2, 3, and 4, confirming the conclusion across various dimensions.

PREREQUISITES
  • Understanding of matrix theory and determinants
  • Familiarity with row reduction techniques
  • Basic knowledge of triangular matrices
  • Experience with mathematical proofs and reasoning
NEXT STEPS
  • Study the properties of determinants in linear algebra
  • Learn about row echelon form and its applications
  • Explore the concept of matrix rank and its implications
  • Investigate alternative methods for calculating determinants, such as cofactor expansion
USEFUL FOR

Students and professionals in mathematics, particularly those studying linear algebra, as well as educators looking for clear explanations of determinant calculations.

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How can I get the determinant of this matrix?

1-n 1 ...1 1
1 1-n ...1 1
. . . .
. . . .
1 1 ... 1 1-n

I think that the answer is 0 but... why?

Thank you.
 
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encomes said:
How can I get the determinant of this matrix?

1-n 1 ...1 1
1 1-n ...1 1
. . . .
. . . .
1 1 ... 1 1-n

I think that the answer is 0 but... why?

Thank you.
Is "n" here the order of the matrix? If so then, by a row reduction to diagonal or triangular form, you can show that the last row becomes all "0"s.
 
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Thx for your reply, but I can't see how u get to this solution.. Can u show me the process?
Thanks again!
 
Try this: add all of the other rows to the first row. What do you get?

(Try it with n= 2, 3, and 4 first.)
 
Alright, so, if I have a determinant with n=4, i add all the rows to the first and i get (4-n +4-n +4-n +4-n), so, it's 0.

Is there any other process?

Thank you.
 

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