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Homework Help: Proofs with matrices

  1. Apr 12, 2007 #1

    daniel_i_l

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    1. The problem statement, all variables and given/known data
    Prove or disprove the following:
    (A is a nxn square matrix)
    a) The vector b is in R^n and all its elements are even integers. If all the elements of the A are integers and det(A) = 2, then the equation Ax = b has a solution with only integer elements
    b) If n is odd and transpose(A) = -A then the equation Ax=0 has only one solution
    c)adj(gammaA) = gamma^n-1adj(A) if gamma is in R.


    2. Relevant equations
    a) Cramer's rule.
    b) det(transpose(A)) = det(A)
    det(-A) = (-1)^ndet(A)
    c) definition of adjoint (transpose of cofactor matrix)


    3. The attempt at a solution

    a)True: according to Cramer's rule each element of the solution is the determinant of A with one coloum replaced by b divided by det(A). Since the determinant can be taken with coloum b the first determinant will be the sum of even numbers (because all of b's elements are even and A's are integers) and that divided by |A|=2 will be an integer.

    b)False: |transpose(A)| = |A|. |-A| = (-1)^nA = -|A| cause n is odd. So 2|A|=0 => |A|=0 and A isn't reversable so it must have a non trivial solution to Ax=0.

    c)True: because each element of adj(gammaA) is +/-1 times a determinant of gamma * (a n-1xn-1 matrix) which equals gamma^n-1 times the determinant of the n-1xn-1 matrix.

    Are the answers right? I'm especially hesitant about c).
    Thanks!
     
    Last edited: Apr 12, 2007
  2. jcsd
  3. Apr 12, 2007 #2

    Dick

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    They seem just fine to me.
     
  4. Apr 12, 2007 #3

    daniel_i_l

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    Gold Member

    Thanks a lot!
     
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