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Can someone check my work? EASY PEASY MATRICES

  1. Apr 30, 2009 #1
    i just need someone to check my awnsers and working XD thanks
    they get harder btw ;)


    1:
    a) Write out the system of linear equations in variables, x,y,z represented by the augmneted matrix

    ( 1 2 0 5)
    ( 3 1 -1 0)
    ( 0 -2 1 2)

    my attempt at 1a:

    x + 2y = 5
    3x + y -z = 0
    -2y + z =2

    1b:
    write down the augmented matrix corresponding to this system of linear equations:

    4a + 2c - d = 8
    4a + b +3c +2d = 0
    b + c + d = 4
    -3a + 8b = 12


    (4 0 2 -1 8)
    (4 1 3 2 0)
    (0 1 1 1 4)
    (-3 8 0 0 12)

    i think that was it?




    2a:
    state whether each of the following matrices is in reduced echelon form. If it is not, then give a reason and say what single row operation is needed to bring it to reduced echelon form.


    I)

    1 0 0 1
    0 1 0 -2
    0 0 2 -5

    not in reduced echelon form
    divide p3 by 2

    II)

    1 1 0 3 0
    0 0 1 2 0
    0 0 1 2 1

    not in reduced echelon form ( don't know how to do this one)
    NOT SURE WHAT SINGLE OPERATION will bring to reduced echelon form

    III)

    1 0 -2 0
    0 1 0 0
    0 0 0 1

    not in reduced echelon form(not sure how to remove the -2)
    NOT SURE WHAT SINGLE OPERATION will bring to reduced echelon form

    IV)

    1 0 2 0 1
    0 0 0 1 2
    0 1 0 0 -3

    not in reduced echelon form
    Swap row three with row two

    V)

    1 0 1 0
    0 1 0 1
    0 0 1 1

    not in reduced echelon form
    row one - row three

    VI)

    0 1 5 0 1
    0 0 0 1 2
    0 0 0 0 0

    not in reduced echelon form (there is a 5 in the way)
    NOT SURE WHAT SINGLE OPERATION will bring to reduced echelon form

    VII)

    1 4 0 1
    0 0 0 0
    0 0 1 3

    not in reduced echelon form(there is a 4 in the way)
    NOT SURE WHAT SINGLE OPERATION will bring to reduced echelon form

    VIII)

    1 0 2 0 3
    0 0 1 0 0
    0 0 0 1 5

    not in reduced echelon form
    row one - (2*row2)




    For the following systems of equations, write down the Augmented matrix and use Gauss reduction as defined in lectures to find it's reduced row echelon form. hence find the solution of the System




    3a:
    x + 3y + z = 4
    2x + 6y -z = -1
    3x +11y +5z =20


    matrix/my solution,
    (1 3 1 4)
    (2 6 -1 -1)p2-2p1
    (3 11 5 20)p3-3p1
    =
    (1 3 1 4)
    (0 0 -3 -9)p2 swap p3
    (0 2 2 8)
    =
    (1 3 1 4)
    (0 2 2 8)p2/2
    (0 0 -3 -9)
    =
    (1 3 1 4)p1-3p2
    (0 1 1 4)
    (0 0 -3 -9)
    =
    (1 0 -2 -8)
    (0 1 1 4)
    (0 0 -3 -9)p3/-3
    =
    (1 0 -2 -8)p1+2p3
    (0 1 1 4)p2-p3
    (0 0 1 3)
    =
    (1 0 0 -2)
    (0 1 0 1)
    (0 0 1 3)
    that is the reduced echelon form
    x = -2
    y = 1
    z = 3



    3b:
    3y+2z = 25
    3x + y + 5z = 29
    x + 2y + 5z = 28

    my solution/matrix
    (0 3 2 25)
    (3 1 5 29)
    (1 2 5 28)p3 swap with p1
    =
    (1 2 5 28)
    (3 1 5 29)p2-3p1
    (0 3 2 25)
    =
    (1 2 5 28)
    (0 -5 -10 -55)p2/-5
    (0 3 2 25)
    =
    (1 2 5 28)p1-2p2
    (0 1 2 11)
    (0 3 2 25)p3-3p2
    =
    (1 0 1 6)
    (0 1 2 11)
    (0 0 -4 -8)p3/-4
    =
    (1 0 1 6)p1-p3
    (0 1 2 11)p2-2p3
    (0 0 1 2)
    =
    (1 0 0 4)
    (0 1 0 7)
    (0 0 1 2)
    this is the reduced echelon form
    x = 4
    y = 7
    z = 2




    4a LAST QUESTION
    while on holiday in europe, bills' daily exspenses were:

    in england, 20 on accomodation, 40 on food 30 on travel
    in france, 60 on accomodation, 30 on food, 20 on travel
    in spain, 40 on accomodation, 35 on food, 35 on travel
    in total he spent 580 on accomodation, 530 on food and 420 on travel


    using e,f and s write down a system of linear equations and solve using gauss reduction to find how many days bill spen tin each country, show all working with clearly labelled reduction steps.

    MY ATTEMPT
    england = x, france = y, spain = z (just to make it simpler)

    20x + 60y + 40z = 580
    40x + 30y + 35z = 530
    30x + 20y + 35z = 420

    matrix
    (20 60 40 580)p1/20
    (40 30 35 530)
    (30 20 35 420)
    =
    (1 3 2 29)
    (40 30 35 530)p2 - 40p1
    (30 20 35 420)p3 - 30p1
    =
    (1 3 2 29)
    (0 -90 -45 -630)p2/-90
    (0 -70 -25 -450)
    =
    (1 3 2 29) p1-3p2
    (0 1 0.5 7)
    (0 -70 -25 -450)p3 + 70p2
    =
    (1 0 0.5 8)
    (0 1 0.5 7)
    (0 0 10 40)p3/10
    =
    (1 0 0.5 8)p1+(.5*p3)
    (0 1 0.5 7)p2+(.5*p3)
    (0 0 1 4)
    =
    (1 0 0 10)
    (0 1 0 9)
    (0 0 1 4)
    this is the reduced echelon form
    x = 10
    y = 9
    z = 4

    10 days in england, 9 in france, 4 in spain

    please check for me and thank you very much XD
     
  2. jcsd
  3. Apr 30, 2009 #2

    Mark44

    Staff: Mentor

    You're more likely to get a response if you don't throw a whole laundry list of problems at us.
     
  4. Apr 30, 2009 #3
    1 a and b are correct
     
  5. Apr 30, 2009 #4
    sorry,
     
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