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Linear algebra - adjoint matrix = 0

  1. Jan 30, 2010 #1
    1. The problem statement, all variables and given/known data

    Ok so the question is:

    Let A be an nxn matrix where n [tex]\geq[/tex] 2. Show A* = 0 (A* is the adjoint matrix, 0 is the zero matrix) if and only if rank(A)[tex]\leq[/tex] n-2.

    2. Relevant equations

    A*:= ((-1)i+jdet(Aij))ij where Aij is the (i,j) minor of A

    3. The attempt at a solution

    I'm really struggling with answering this.

    If rank(A) [tex]\leq[/tex] n-2 then A contains at least 2 dependent rows. Does this automatically make A* = 0?

    If A*=0 then (-1)i+jdet(Aij) = 0 for all i and j. So det(Aij) = 0 for all i and j. Having a zero determinant implies dependence?
  2. jcsd
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