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Math Help - Degrees of Freedom in Matrix?

  1. #1
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    Degrees of Freedom in Matrix?

    I have a matrix:

    1 a 2a 0 1
    0 0 1 -(a+1)/(3a-2) -1/(3a-2)
    0 0 0 a^2 - a-2 a^2-3a-4

    What values of "a" create infinite solutions with 1 degree of freedom?
    What values of "a" create infinite solutions with 2 degree of freedom?
    What values of "a" create infinite solutions with 3 degree of freedom?

    My problem is...i'm not to sure what degrees of freedom mean? Google and my book both come up pretty short on that definition.

    Can somoene please explain in laymans terms what that means?

    I know the nullity = 1 or 2 depending if a = -1 or not. "a" can never be -2

    I guess the value for a = -2 for 2 degrees of freedom and a != -2 for 1 degree of freedom?
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  2. #2
    A Plied Mathematician
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    I think degrees of freedom in solving a matrix system work like this: suppose you have an underdetermined matrix (not a contradictory system with two different quantities equal to the same quantity). When you end up solving for x, you get, usually, a vector plus t times another vector. That would be one degree of freedom (t), which is what you get if you have one fewer equations than unknowns. If you have a consistent system with two fewer equations than unknowns, you would get a vector plus t times a vector plus s times another vector. That would be two degrees of freedom. So, to solve your problem, you need to find out which values of a give you redundant lines in your matrix.
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  3. #3
    MHF Contributor

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    Part of the problem is that what you have said makes no sense! A matrix does not have "solutions" to begin with so it makes no sense to talk of a value of a that "creates infinite solutions".

    I think what you are talking about is a system of three linear equations and the matrix is the "augmented matrix" for that system. It is the system of equations that may have infinite solutions.

    Row reduce the system so that you have a function of a only in the last row and last column. For some a, the entry in the third column, last row, is 0. If the entry in the fourth column, last row, is not also 0, there are NO solutions. If it is 0 for the same value of a that makes the third column, last row 0, then there will be infinite solutions to the system of equations.
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