1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Linear Algebra determine the conditions

  1. Aug 29, 2010 #1
    1. The problem statement, all variables and given/known data

    determine the conditions on a b c and d such that the matrix
    a b
    c d
    will be row equivalent to the given matrix:
    1 0
    0 1

    and

    1 0
    0 0

    2. Relevant equations


    3. The attempt at a solution

    I have no idea what its asking, I mean on no. 1:
    a = 1 b = 0 c = 0 and d =1
    but what does that mean? I tried taking the resultant

    for the first one would be 1(1) - 0(0) = 1

    which would be ad - bc = 1 ??
     
  2. jcsd
  3. Aug 29, 2010 #2

    Mark44

    Staff: Mentor

    This looks to me like two separate problems, or one problem with two parts.
    Determine the conditions on a b c and d such that the matrix
    a b
    c d
    will be row equivalent to the given matrix:
    a)
    1 0
    0 1

    b)
    1 0
    0 0

    By "resultant" I think you mean determinant, and you're on the right track with that approach.
     
  4. Aug 29, 2010 #3
    yes, they are 2 problems, and yes I meant the determinant :P
    still if im on the right approach. . .thats as far as I can go. . .is that the answer?
    a)
    ad - bc = 1


    b)
    ad - bc = 0
     
    Last edited: Aug 29, 2010
  5. Aug 29, 2010 #4

    Mark44

    Staff: Mentor

    You should start a new thread for the other problem.

    For the original problem, suppose the matrix is
    [2 1]
    [3 0]

    Will it be row-equivalent to the identity matrix or to the other one (the one that has all zeroes except for the upper left entry)?
     
  6. Aug 29, 2010 #5
    2(0) - 3(1) = -3 . . .Im guessing no? (its not zero nor 1)
    or if you do some opperations
    -r1 + r2 =
    [1 -1]
    [3 0]
    and r_3 - 3r1

    [1 -1]
    [0 3]
    divide by 3 on row 2
    [1 -1]
    [0 1]

    add row_1 + row_2
    [1 0]
    [0 1]
    what I dont understand is what
    row equivalency is? unless it means that its like "multiples" (or row-operation-wise) of each other?

    if so then what you mentioned is row equivalent to a) ??
    still how would I write the answer?
     
  7. Aug 30, 2010 #6

    HallsofIvy

    User Avatar
    Science Advisor

    Okay, now do the same that to the matrix given: actually row-reduce the given matrix!

    From
    [tex]\begin{bmatrix}a & b \\ c & d\end{bmatrix}[/tex]
    as long as [itex]a\ne 0[/itex] we can divide the first row by a, then subtract c times the first row from the second to get
    [tex]\begin{bmatrix}1 & \frac{b}{a} \\ 0 & d- \frac{bc}{a}\end{bmatrix}[/tex]

    If [itex]d- bc/a\ne 0[/itex] (which is the same as saying det= ad- bc= 0), we have the second form. If not, we can divide the second row by that and get the first form.

    If a= 0, we can swap the rows, getting
    [tex]\begin{bmatrix}c & d \\ a & b\end{bmatrix}[/tex]
    Now what?
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook