1. Not finding help here? Sign up for a free 30min 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!

3x2 matrices in RREF

  1. Apr 21, 2015 #1
    1. The problem statement, all variables and given/known data
    How many 3x2 matrices are there in RREF form?

    2. Relevant equations


    3. The attempt at a solution
    I counted 5, but the solutoin in my book says 4.

    0 0
    0 0
    0 0


    1 0
    0 1
    0 0

    1 1
    0 0
    0 0


    0 1
    0 0
    0 0

    1 0
    0 0
    0 0
     
  2. jcsd
  3. Apr 21, 2015 #2

    Mark44

    Staff: Mentor

    Think about these matrices as a system of three equations in two unknowns, x and y.
    ax + by = 0
    cx + dy = 0
    ex + fy = 0
    This system of equations represents three lines in the plane. The 2nd reduced matrix you show above says that all three lines intersect at a single point, the origin - (0, 0). What do the other matrices represent geometrically?
     
  4. Apr 21, 2015 #3
    The 0 zero matrix represents of all of the plane. The 3rd, 4th, and 5th matrix represent some line on the plane.


    Aren't all 5 of those matrices in RREF form though?
     
  5. Apr 21, 2015 #4
    Oh wait I see.

    1 0
    0 0
    0 0


    1 1
    0 0
    0 0

    are essentially the same since that second element (top right) can be anything (e.g., 0, 1, .....)
     
  6. Apr 21, 2015 #5

    Mark44

    Staff: Mentor

    No, I don't think so. The first matrix says x = 0, so all three lines lie along the y axis. The second matrix says x + y = 0, or y = -x, so all three lines lie along this line.
     
  7. Apr 21, 2015 #6

    Mark44

    Staff: Mentor

    What would the system have to look like so that you ended up with the first matrix (the one with all zeroes)?
     
  8. Apr 21, 2015 #7
    Wouldn't the system just be zero?

    I can't think of any matrices that would row reduce to 0 besides 0 itself.
     
  9. Apr 21, 2015 #8

    Mark44

    Staff: Mentor

    The system wouldn't be 0, it would have to look like this, wouldn't it?
    0x + 0y = 0
    0x + 0y = 0
    0x + 0y = 0
    My thought is that they aren't considering this as a legitimate system of equations.
     
  10. Apr 21, 2015 #9
    Oh, hmmm. I think my book considers the 0 matrix as in RREF.

    And I'm a little confused with the semantics here. Why is the system not zero? All I see are zeros.

    I think my post #4 is correct, since we can put any # in that top right element:
    i.e.,
    1 c
    0 0
    0 0
    where c can be anything.
     
  11. Apr 21, 2015 #10

    Mark44

    Staff: Mentor

    One other thought - what you show as
    1 1
    0 0
    0 0

    should probably look more like
    a b
    0 0
    0 0
    Edit:
    1 c
    0 0
    0 0
    would work as well.


    Your (1 1) version provides for the possibility that the three lines all lie on the graph of y = -x. My version allows for all possible non-vertical, non-horizontal lines.

    For me, it's helpful to look at the geometry -- the system of three equations in two variables represents three lines that all go through (0, 0), since each equation is of the form ax + by = 0. Since they all go through (0, 0) it's not possible that we have a system where one or two lines are parallel to the third, hence no solution. So we have to have one of the following:

    All three lines are vertical -- x = 0, your 4th matrix
    All three lines are vertical -- y = 0, your 5th matrix
    All three lines intersect at the origin -- your 2nd matrix
    All three lines lie on the same non-vertical, non-horizontal line -- my matrix above

    As far as your matrix of all zeroes, that's a legitimate matrix, but I don't see how it could be the result of any non-trivial system of equations.
     
  12. Apr 21, 2015 #11

    Mark44

    Staff: Mentor

    Be careful to distinguish between the number 0 and the ##\mathcal{0}## matrix. Here it should be understood that we're talking about matrices, all of whose elements are zero.

    Response to your question below: A system consists of one or more equations. Equations are different from numbers or matrices. There is no notation that I know of to represent a "zero" system. What I'm saying is, don't describe a system as "zero", but I guess you could talk about a system of equations where all of the coefficients are zero.


     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted