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Some linear algebra problems i need help with

  1. Mar 24, 2009 #1
    Let A and B be nxn matrices.
    1. Suppose that AB=AC and det A does not equal 0. Show that B=C

    2. Show that A is nonsingular if and only if A transpose is nonsingular.

    3. Show that det AB = det BA.

    4. Show that det AB = 0 if and only if det A=0 or det B=0

    5. Show that if AB= -BA and n is odd, then A or B is singular.

    6. Show that det A*Atranspose is greater than equal to 0

    7. Show that det A*Btranspose = det Atranspose* det B

    8. Let A be nxn skew-symmetric matrix. If n is odd, show that det A=0

    9. Show that 3x3 vandermonde matrix has a determinant equal to (a-b)(b-c)(c-a) The matrix is
    [1 1 1
    a b c
    a^2 b^2 c^2]
    Thank you.
     
  2. jcsd
  3. Mar 24, 2009 #2
    1. [tex]AB=AC\Rightarrow A(B-C)=0[/tex]
    if A is regular, then there are all pivots nonzero. Then only one way is to satisfy that equation, so [tex]B-C=0\Rightarrow B=C\qquad\square[/tex]
    2. Take any matrix in echelon form, with some pivots. If one of them is zero, then also traspose has a zero pivot. Then A is singular and A transpose is singular.
    3. [tex]\det A\det B=\det A\det B\Rightarrow\det A\det B=\det B\det A\Rightarrow\det AB=\det BA[/tex]
     
  4. Mar 24, 2009 #3

    HallsofIvy

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    I would really like to see some work on your part. If nothing else it would help to determine what kind of hints would help you. For example, I can see three different ways to do problem 1 but I don't know which way would be best for you.
     
  5. Mar 24, 2009 #4
    4. Show that det AB = 0 if and only if det A=0 or det B=0
    well i know that I have to show two parts for this one
    part 1 that assume that det AB=0 then show that det A=0 or B=0
    part 2 assume that det A=0 or B=0 then show that det AB = 0
    but I have hard time coming up with a good organization and details for this kinds of problem.


    5. Show that if AB= -BA and n is odd, then A or B is singular.
    i don't have any clue how to start this one... please give me any hints..

    6. Show that det A*Atranspose is greater than equal to 0
    hmmm i have no clue...
    7. Show that det A*Btranspose = det Atranspose* det B

    8. Let A be nxn skew-symmetric matrix. If n is odd, show that det A=0

    9. Show that 3x3 vandermonde matrix has a determinant equal to (a-b)(b-c)(c-a) The matrix is
    [1 1 1
    a b c
    a^2 b^2 c^2]

    when i found the det for this.. I got bc^2+ca^2+ab^2-ba^2-cb^2-ac^2.. i don't know if this is right.. and don't know where to go from there...

    I am trying my best and if anyone could give me some type of hints or help me through these problems... that would be great...
    Thanks
     
  6. Mar 25, 2009 #5
    4. You can use the result of a theorem( i don't know whether they expect you to prove it as well or not)

    det(AB)=det(A)det(B).

    Now if you suppose that det(AB)=0=> det(A)det(B)=0=>....? and vice-versa

    5. is n supposed to be the dimension of the matrices A and B?
     
  7. Mar 25, 2009 #6
    for num 5. yes n is suppose to be the demention of the matrices, so they are square matrices.
     
  8. Mar 25, 2009 #7
    Also you can, probbaly use another result:

    det(A)=det(A^T)

    A^T=A transpoze. This will help you for 6 and 7..
     
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