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

  1. Sep 18, 2009 #1
    Suppose we have a system of 2 equations in 2 unknowns x,y:

    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]
    [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. Sep 20, 2009 #2
    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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