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Math Help - Bx=b showing consistency

  1. #1
    n22
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    Bx=b showing consistency

    Show that the system of equations Bx=b is consistent for all bεℝ

    B=[1 2 0 1;1 0 1 -1;0 -1 3 -4]

    Please note that
    ; means row; This is a 3 by3 matrix
    Often in answers I find that they say let ( x y z ) be an element of R^3 and then they put it in the matrix(on the Rhs as a solution) and do reduced row echelon...Is that necessary here? why is that necessary in other cases?
    (its possible that I dont need to do RREs but lets say i did ..is this method correct?)
    Anyway heres what I have done.
    I would appreciate feedback.THanks.
    Please see attachments.
    Bx=b showing consistency-dsc_0103-1.jpgBx=b showing consistency-dsc_0104-1-1-1.jpg



    Attached Thumbnails Attached Thumbnails Bx=b showing consistency-dsc_0104-1-1-1-1-.jpg  
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  2. #2
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    Re: Bx=b showing consistency

    Hey n22.

    Hint: Reduce the matrix and if you get a rank of less than 3 (i.e. zeroes in the first three columns for a row), then see if the fourth corresponding value is zero or non-zero.

    The idea is that if 0x + 0y + 0z != 0, then you know that you have an inconsistent system.
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  3. #3
    n22
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    Re: Bx=b showing consistency

    Hi Chiro,
    Finally I end up with this matrix: Bx=b showing consistency-matrix-.jpg

    [x y z]=t[3 -2 -2] where t is some parameter which is an element of R..please not this is a column vector..if there is a parameter t doesnt that mean that there are an infinite number of solutions ?
    Rank=dim(colA)=no leading ones =3
    What does this mean for the matrix???
    i guess this means its consistent then ....since there is a solution for each leading one


    Quote Originally Posted by chiro View Post
    Hey n22.

    Hint: Reduce the matrix and if you get a rank of less than 3 (i.e. zeroes in the first three columns for a row), then see if the fourth corresponding value is zero or non-zero.

    The idea is that if 0x + 0y + 0z != 0, then you know that you have an inconsistent system.
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  4. #4
    MHF Contributor
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    Re: Bx=b showing consistency

    Since you are reducing in terms of free parameters x, y, and z, then what did you get in terms of x,y,z for the final column? (You have all zeroes but they should be functions of x, y, and z).

    If you want to check the answer, I suggest you get something like Maple or Mathematica (Maple is good for this problem).
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