Hey there! Sign in to join this conversationNew here? Join for free
    • Thread Starter
    Offline

    15
    ReputationRep:
    Let A \in M_n (\mathbb{Z} ) be a square matrix with integer entries.

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

    Prove that det( A)=\pm 1.
    Help? :cry2:
    Offline

    2
    ReputationRep:
    A matrix is invertible if and only if its determinant is invertible. (Easy proof.) Which integers have a multiplicative inverse in the integers?
    • Thread Starter
    Offline

    15
    ReputationRep:
    (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:
    • Thread Starter
    Offline

    15
    ReputationRep:
    Bump
    Offline

    16
    ReputationRep:
    You don't need to prove the if, just the only if. Remember, \det A B = \det A \det B
    • Thread Starter
    Offline

    15
    ReputationRep:
    (Original post by SimonM)
    You don't need to prove the if, just the only if. Remember, \det A B = \det A \det B
    How would I apply detAB=detAdetB to this proof/question?

    Where does the matrix B come from?

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

    Thanks.
    Offline

    16
    ReputationRep:
    Well, aside from A, what other matrix do you have?
    • Thread Starter
    Offline

    15
    ReputationRep:
    Is this correct?

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


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


    detI=1, so,


    detA.detA^{-1}=1


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

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


    Also, can you find an example of a 3x3 matrix A with the above properties such that A has all nonzero entries?

    Thanks
    • Thread Starter
    Offline

    15
    ReputationRep:
    Bump
    Offline

    2
    ReputationRep:
    (Original post by hollywoodbudgie)
    Since the matrices in this case only have integer values, the only possible determinants are

    \pm 1, and detA=detA^{-1}
    Yes, but your second equation isn't true in general. (It is true in this very special case because \det A = \pm 1.)

    Also, can you find an example of a 3x3 matrix A with the above properties such that A has all nonzero entries?
    Yes, such matrices exist. (I found one on my second attempt by plugging in random positive integers.)
 
 
 
  • See more of what you like on The Student Room

    You can personalise what you see on TSR. Tell us a little about yourself to get started.

  • Poll
    Did TEF Bronze Award affect your UCAS choices?
    Useful resources

    Make your revision easier

    Maths

    Maths Forum posting guidelines

    Not sure where to post? Read the updated guidelines here

    Equations

    How to use LaTex

    Writing equations the easy way

    Student revising

    Study habits of A* students

    Top tips from students who have already aced their exams

    Study Planner

    Create your own Study Planner

    Never miss a deadline again

    Polling station sign

    Thinking about a maths degree?

    Chat with other maths applicants

    Can you help? Study help unanswered threads

    Groups associated with this forum:

    View associated groups
  • See more of what you like on The Student Room

    You can personalise what you see on TSR. Tell us a little about yourself to get started.

  • The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

    Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

    Quick reply
    Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.