Solving Exercise 12 from Hoffman & Kunze's Linear Algebra

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SUMMARY

The discussion focuses on solving Exercise 12 from Hoffman and Kunze's "Linear Algebra," specifically regarding the invertibility of a matrix and finding its inverse. Participants suggest using example 16 from the book to derive the inverse through matrix multiplication. The goal is to demonstrate that the matrix is invertible and that its inverse contains integer entries, rather than providing a full proof of the inverse itself. The conversation highlights the importance of elementary methods in linear algebra problem-solving.

PREREQUISITES
  • Familiarity with matrix multiplication
  • Understanding of matrix invertibility
  • Knowledge of integer matrices
  • Basic concepts from Hoffman and Kunze's "Linear Algebra"
NEXT STEPS
  • Review example 16 from Hoffman and Kunze's "Linear Algebra" for practical applications
  • Study the properties of Hilbert matrices and their inverses
  • Explore elementary proofs of matrix invertibility
  • Practice matrix multiplication techniques for verifying inverses
USEFUL FOR

Students of linear algebra, educators teaching matrix theory, and anyone seeking to deepen their understanding of matrix invertibility and practical problem-solving techniques in linear algebra.

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Hi all.

From Hoffman and Kunze's book Linear Algebra, page 27, exercise 12:

attachment.php?attachmentid=44672&stc=1&d=1330812572.jpg


What's the most elementary way to solve this?

Any hint is appreciated.
 

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Does example 16 calculate several small examples of these? The easiest way might be to use the examples to guess what the inverse is and then just prove through matrix multiplication that it works
 
Office_Shredder said:
Does example 16 calculate several small examples of these? The easiest way might be to use the examples to guess what the inverse is and then just prove through matrix multiplication that it works

If you can't guess, Google is your friend.
 
Thanks for the replies.

I've searched by Google (actually the book doesn't mention the name of the matrix), and all I found is some stronger results (like in Wikipedia: http://en.wikipedia.org/wiki/Hilbert_matrix) with not enough elementary proofs. I don't need to prove what the inverse matrix is, just need to prove it is invertible, and the inverse has integer entries.

Here is the example 16:
attachment.php?attachmentid=44699&stc=1&d=1330851101.jpg
 

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