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Semantic/Logical Issue

  1. Jan 8, 2017 #1
    1. The problem statement, all variables and given/known data
    Consider the system ##AX=0##, where
    $$A = \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix}$$
    is a 2x2 matrix over the field F. Prove that if ##ad-bc=0## and some entry of ##A## is different from ##0##, then there is a solution ##(p,q)## such that ##(x,y)## is a solution if and only if there is some scalar ##t## such that ##x=pt## and ##y = qt##.

    2. Relevant equations


    3. The attempt at a solution
    Am I asked to find the vector ##(p,q)## for which the statement "##(x,y)## is a solution if and only if there is some scalar ##t## such that ##x=pt## and ##y = qt##" holds, or am I assuming that I have such a solution ##(p,q)## and proving that this statement?
     
  2. jcsd
  3. Jan 8, 2017 #2

    stevendaryl

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    Staff Emeritus
    Science Advisor

    Well, the problem doesn't ask you to find [itex](p,q)[/itex], only to show that if [itex](p,q)[/itex] is a solution, and [itex](x,y)[/itex] is a second solution, then there is some [itex]t[/itex] such that [itex]x=pt[/itex] and [itex]y=qt[/itex].

    For a 2x2 matrix, it's easy enough to convert [itex]\left( \begin{array}\\ a & b \\ c & d \end{array} \right) \left( \begin{array} \\ x \\ y \end{array} \right) = 0[/itex] into two ordinary (non-matrix) equations for [itex]x[/itex] and [itex]y[/itex] to see how the possible values of [itex]x[/itex] and [itex]y[/itex] must be related.
     
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