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Homework Help: Linear algebra proof problem

  1. Mar 6, 2009 #1
    1. The problem statement, all variables and given/known data
    Suppose that AX = O is a homogeneous system of n equations in n variables. If the system (A^2)X = O has a nontrivial solution, show that AX = O has a nontrivial solution.

    2. Relevant equations
    Reduced row echelon form definition, matrix multiplication, etc.

    3. The attempt at a solution
    This looks like it would be easier to prove the contrapositive:
    If AX = O does not have a nontrivial solution, then (A^2)X does not have a nontrivial solution.

    However I'm not sure how to solve this.
    If AX = O does not have a nontrivial solution, then the bottom row of A in reduced row echelon form is not all 0's. Should I use that to prove the bottom of A^2 in reduced row echelon form is not all 0's? Because I'm having trouble with that. Or maybe there is a different way to prove this problem.
  2. jcsd
  3. Mar 6, 2009 #2


    Staff: Mentor

    From the given information, the matrix A is n x n. You're also given that A2x = 0 has a nontrivial solution, which has implications about the value of the determinant |A2|. Is that enough to get you started?
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