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If det(A)!=0, Ac=0 has only trivial solution

  1. Oct 29, 2011 #1

    mzh

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    1. The problem statement, all variables and given/known data
    Show that the equation Ac = 0, where A is a NxN matrix and c is a column matrix with elements c_i, i=1..N can have a non-trivial solution (c != 0) only when det(A) = 0.


    2. Relevant equations
    inv(A) does not exist when det(A) = 0.


    3. The attempt at a solution
    If det(A) != 0, we can form inv(A). Then we can write
    inv(A)*A*c=inv(A)*0
    and so
    c=0,
    and so we arrive at the trivial solution. On the contraire, if det(A)=0, we can not form the inverse, and so we can not write
    inv(A)*A*c=inv(A)*0, because inv(A) does not exist.
    So, there must be other solutions.

    Does that make sense? How do I write this in a more formal way?

    Thank you for any suggestions
     
  2. jcsd
  3. Oct 29, 2011 #2
    What you've said is equivalent to
    To prove this, it is logically equivalent to show the contrapositive. Namely
    This is what you have done in the first part of your post, so you don't need the second part.
     
  4. Oct 29, 2011 #3

    mzh

    User Avatar

    Hey Kreizhn, thanks for your feedback. Ok, i see I'm saying two time the same thing. I was not sure if the first part would allow to conclude the second part. So your second argument is sufficient as a proof, i understand. And this does not require any more statements to be complete (for non-mathematicians).
     
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