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Matrices

Let A∈Mn(Z)A \in M_n (\mathbb{Z} ) be a square matrix with integer entries.

Suppose that AA is invertible and its inverse A−1A^{-1} also has integer entries.

Prove that det(A)=±1det( A)=\pm 1.


Help? :cry2:
Reply 1
Avatar for Zhen Lin
Zhen Lin
12
A matrix is invertible if and only if its determinant is invertible. (Easy proof.) Which integers have a multiplicative inverse in the integers?
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. :top:
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Reply 4
Avatar for SimonM
SimonM
18
You don't need to prove the if, just the only if. Remember, det⁡AB=det⁡Adet⁡B\det A B = \det A \det B
Original post by SimonM
You don't need to prove the if, just the only if. Remember, det⁡AB=det⁡Adet⁡B\det A B = \det A \det B


How would I apply detAB=detAdetBdetAB=detAdetB to this proof/question?

Where does the matrix BB come from?

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

Thanks.
(edited 13 years ago)
Reply 6
Avatar for SimonM
SimonM
18
Well, aside from A, what other matrix do you have?
Is this correct?

We know A.A−1=IA.A^{-1}=I and det(AB)=detA.detBdet(AB)=detA.detB


So, detA.detA−1=detIdetA.detA^{-1}=detI


detI=1detI=1, so,


detA.detA−1=1detA.detA^{-1}=1


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

±1\pm 1, and detA=detA−1detA=detA^{-1}


Also, can you find an example of a 3x33x3 matrix AA with the above properties such that AA has all nonzero entries?

Thanks
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Reply 9
Avatar for Zhen Lin
Zhen Lin
12
Original post by hollywoodbudgie
Since the matrices in this case only have integer values, the only possible determinants are

±1\pm 1, and detA=detA−1detA=detA^{-1}


Yes, but your second equation isn't true in general. (It is true in this very special case because det⁡A=±1\det A = \pm 1.)

Also, can you find an example of a 3x33x3 matrix AA with the above properties such that AA 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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