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# Matrices Watch

1. Let be a square matrix with integer entries.

Suppose that is invertible and its inverse also has integer entries.

Prove that .
Help?
2. A matrix is invertible if and only if its determinant is invertible. (Easy proof.) Which integers have a multiplicative inverse in the integers?
3. (Original post by Zhen Lin)
A matrix is invertible if and only if its determinant is invertible. (Easy proof.) Which integers have a multiplicative inverse in the integers?
Sorry, I don't understand. How would I go about doing this 'easy proof'?

Thanks.
4. Bump
5. You don't need to prove the if, just the only if. Remember,
6. (Original post by SimonM)
You don't need to prove the if, just the only if. Remember,
How would I apply to this proof/question?

Where does the matrix come from?

Sorry, I just can't seem to figure where to apply it.

Thanks.
7. Well, aside from A, what other matrix do you have?
8. Is this correct?

We know and

So,

, so,

Since the matrices in this case only have integer values, the only possible determinants are

, and

Also, can you find an example of a matrix with the above properties such that has all nonzero entries?

Thanks
9. Bump
10. (Original post by hollywoodbudgie)
Since the matrices in this case only have integer values, the only possible determinants are

, and
Yes, but your second equation isn't true in general. (It is true in this very special case because .)

Also, can you find an example of a matrix with the above properties such that has all nonzero entries?
Yes, such matrices exist. (I found one on my second attempt by plugging in random positive integers.)

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