Solving Exercise 12 from Hoffman & Kunze's Linear Algebra

[asmani]

Hi all.

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

What's the most elementary way to solve this?

Any hint is appreciated.

 

[Office_Shredder]

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

 

[AlephZero]

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.

 

[asmani]

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: