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True / False

  1. Apr 20, 2010 #1

    Answer as T or F:

    1) Every linear system consisting of 3 equations in 4 unknowns has infintely many solutions.
    2) If A and B are 3 x 3 matrices , then det(AB - A (B^T) ) = 0
    3) If A and B are n x n matrices, nonsingular matrices and AB=BA, then A(B^-1) = (B^-1)A
    4) If A is a singular n x n matrix, then Aadj(A)=0

    For (1):
    I think its true
    since # of columns > # of rows
    so we will have recall a parameter
    and this means we will a infinitely many solutions

    For (2):
    I do not know how to do it =(

    For (3):
    I got the answer, its true
    but how ?

    For (4):
    I completely stopped here :/

    Any help please?

    this is not for my homework
    I swear
    am solving these for fun
  2. jcsd
  3. Apr 20, 2010 #2


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    A hint for the second one: factor out A and use the rule det(XY)=det(X)det(Y). Then, can you conclude something about det(B - B^T)? What are the diagonal elements of that matrix? What is the general element of that matrix? Use the rule of Sarrus to calculate the determinant.
  4. Apr 20, 2010 #3


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    A hint for the third one: multiply the left side of the equation AB = BA by B^-1.
  5. Apr 21, 2010 #4


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    A hint for the fourth one: take an element cij from the matrix C = Aadj(A) and write down what it equals. Can you conclude something from that?
  6. Apr 21, 2010 #5
    Correct me if I'm wrong, but for (1), I believe the answer is False. While you are correct most of the time, you have to consider the situation when the system might have 0 solutions. In general - if r is the rank of the matrix - r < n and r < m implies the system will have 0 or an infinite amount of solutions. Thus, not every system described will have an infinite amount of solutions.
    Last edited by a moderator: Apr 24, 2010
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