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Find vlaues of k so matrix has no solutions, what if u change the position of k?

  1. Dec 11, 2005 #1
    Hello everyone! I have a problem, it says Find hte values of k for which the matrix sysem Ax = b has no solutions. I got the answer for this one,
    A =
    6 2 2
    -1 2 3
    2 -6 k

    b =

    I row reduced and ending up gettting:
    6 3 0
    0 -1 2
    0 0 14-k
    so k = 14, will give no solutions.

    After that he says find:
    Image space, nullspace, col space and row space, what are there dimensions, i also did this correctly.

    But what happens if he changed the problem and said, Find the values of k for which the matrix system Ax = b has infitinatly many solutions or 1 unquie solution, then what would i do different? Thanks! I'm just trying to get all angles of this problem so i don't mess it up on the exam.

    Also what if he puts k in the b vector insteed of the matrix A?
    b =
  2. jcsd
  3. Dec 11, 2005 #2

    matt grime

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    Science Advisor
    Homework Helper

    that depends on A, doesn't it. If A is invertible (cols linearly independent) then there is exactly one unique solution of Ax=b. assume A is singular. then if there exists one solution then there are infinitely many, so look if b is in the image.

    depends on the A, same comments apply.
  4. Dec 11, 2005 #3
    thanks for the responce, To see if A is invertiable, and to find the value of k in which there is exactly 1 unique solution, what would i do to the matrix, is what i was asking, like to find no solutions i row reduced until i got an expression of k, i got k -14 = 0;
    so if k = 14, u got no solutions, say he changed the directions and said, find k so that there is 1 uqniue solution, then would it be like
    k - 14 = ? thats what i don't get
  5. Dec 11, 2005 #4

    matt grime

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    Science Advisor
    Homework Helper

    I told you the answer to that: if the cols are linearly independent then there is a unique solution, and the cols are linearly indepndent exactly when k-14 doesn'te equal zero. you found k-14=0 for a reason, remember. something that is not linearly dependent is linearly independent, it's a binary option. and you know linearly dependent is equivalent to k=14=0....
  6. Dec 11, 2005 #5
    ohhh my bad, thanks!!! everything is running together in my mind hah sorrry.
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