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Determine values of h - augmented matrix

  1. Feb 21, 2013 #1
    1. The problem statement, all variables and given/known data

    YnEsHEY.png

    2. Relevant equations



    3. The attempt at a solution

    Tried to get it on reduced echelon form, but I haven't done problems like this before so I don't know what I'm supposed to do.



    Thanks!
     
  2. jcsd
  3. Feb 21, 2013 #2
    What have you tried? Show us the row operations you've done so we can work from there.
     
  4. Feb 21, 2013 #3
    The thing is I don't know where to start. By multiplying with -4 and adding to second row, I have:

    1 5 -4
    0 20-h 0

    And this doesn't really tell me anything, as I have never done matrix with variables before.
     
  5. Feb 21, 2013 #4

    SteamKing

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    Ignore the augmented -column.

    What condition must hold with the matrix of coefficients in order for the system to be consistent? (hint: it involves a determinant)
     
  6. Feb 21, 2013 #5
    You didn't calculate that properly. Try again.
     
  7. Feb 21, 2013 #6
    Woops, I meant -20+h.
     
  8. Feb 21, 2013 #7
    Yes that was the answer I was looking for, but in this case, it doesn't matter since you could just multiply by -1. So,

    Rewriting the last row in terms of variables, we have:

    ##0*x + (h - 20)*y = 0##
    ##(h - 20)*y = 0##

    Recall that ##a*b = 0## if and only if ##a = 0## or ##b = 0##. One case is that ##y = 0## and that would yield a consistent solution. What must be the second one?
     
  9. Feb 21, 2013 #8
    So if h is 20 the solution would also be consistent?
     
  10. Feb 21, 2013 #9
    Yup, that's correct. Notice that plugging in ##h = 20##, and then row reducing, we get a row of only ##0s## which is consistent. If ##h## was anything but 20, we would have the 2nd row look like: ##0 \ r \ | \ 0## which is inconsistent for any ##r \in \mathbb{R}, r \not = 0##.
     
  11. Feb 21, 2013 #10
    Alright, that made sense. Thanks for the help :)
     
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