System of linear equations-unique, infinitely many, or no solutions

  1. Suppose we have a system of 2 equations in 2 unknowns x,y:
    a1x+b1y=c1
    a2x+b2y=c2

    If the determinant of
    [a1 b1
    a2 b2]
    is nonzero, then the solution to the system exists and is unique. [I am OK with this]

    If the determinant of
    [a1 b1
    a2 b2]
    is zero,
    this does not distinguish between the cases of no solution and infinitely many solutions.

    To gain some insight, we need to further check the determinatnts of
    [a1 c1
    a2 c2]
    and
    [b1 c1
    b2 c2]
    If both determinants are zero, then the system has infinitely many soltuions.
    If at least one of the two determinants are nonzero, then the system has no solution.

    ===================================

    Is the bolded part correct or not? I don't see why it is true. How can we prove it?


    Thanks for any help!
     
    Last edited: Sep 18, 2009
  2. jcsd
  3. Can someone "confirm or disprove" the bolded part, please?

    [my PDE book seems to be applying these ideas from linear algebra to study the solvability of initial value problems for quasilinear partial differential equations, but I can't find those results in my linear algebra textbooks other than the result "det(A) is not 0 <=> solution exists and is unique"]
     
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