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Describe All the solutions to: AX=0 (Square Matrices, A≠0)

  1. Oct 15, 2012 #1
    1. The problem statement, all variables and given/known data
    A and X are square matrices. A≠0

    Describe all solutions to:
    AX=0

    2. Relevant equations
    3. The attempt at a solution
    X[itex]_{i}[/itex] is some solution.
    Solutions:
    • X=0
    • k*X[itex]_{i}[/itex]
    • ƩX[itex]_{i}[/itex]

    Looking for other solutions:
    Let there be B: AB=BA
    AX=0 ⇔ BAX=B0 ⇔ ABX=0

    New solution: BX

    So maybe all solutions are describable as:
    BX=0, AB=BA

    But this needs proof! (and may allow for further refining of the solution)

    Other info:
    If there exists X[itex]_{i}[/itex]≠0 => A[itex]^{-1}[/itex] doesn't exist. (X=A[itex]^{-1}[/itex]0=0 False)

    BX[itex]_{1}[/itex]=X[itex]_{2}[/itex] ⇔ X[itex]_{1}[/itex]=B[itex]^{-1}[/itex]X[itex]_{2}[/itex]
    What if B has no inverse? Then you've gotta state your solution with X[itex]_{2}[/itex]. But this may keep up, so is there any mother matrix from which these sprout? I think I found B without inverse.
     
  2. jcsd
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