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Kernels and determinants of a matrix

  1. Dec 15, 2013 #1
    I read that an equation of the form Ax=0 has a solution iff the matrix A has non-trivial Kernel, which makes sense as if A had trivial kernel then x would be trivial as well, meaning that only the x={0} solution would exist, right?

    Secondly, I read that in order for A to have a non-trivial kernel, we need detA=0. Why is this so?
     
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  3. Dec 15, 2013 #2

    jgens

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    This is more or less correct. I am not sure what the proper terminology is here, but it might be more proper to say something like:
    An equation of the form Ax = 0 has a non-trivial solution if and only if the matrix A has non-trivial kernel.
    I never really learned matrix algebra so maybe the x = 0 solution does not count or something, but it seems like you should add the non-trivial caveat for clarity.

    There are several ways to look at this. Perhaps the simplest (although slightly unenlightening) way to see this is what follows: The determinant is multiplicative, so if A is invertible, then (det A)(det A-1) = 1 and this guarantees that neither of those guys can be zero. On the other hand if det A ≠ 0 then one can construct an inverse matrix. Just multiply the adjugate by (det A)-1 and you have your inverse.
     
  4. Dec 15, 2013 #3
    Although it makes sense, what you are saying sounds like det A≠0, whereas I was looking for det A=0.

    How does your observation relate to my question about "requiring det A=0 in order to have a non-trivial kernel"?
     
  5. Dec 15, 2013 #4

    jgens

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    What happens in the det A = 0 case can be deduced from the det A ≠ 0 case. If you put some thought into, then I am sure you can figure it out.

    It relates in a fairly obvious way.
     
  6. Dec 15, 2013 #5

    AlephZero

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    You can interpret the product Ax as the sum of (the elements of x) times (the column vectors of A).

    So, if Ax = 0 and x ≠ 0, the column vectors of A are linearly dependent, and therefore det A = 0.
     
  7. Dec 15, 2013 #6

    HallsofIvy

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    Actually this is NOT true. "Ax= 0" always has a solution: x= 0. It has non-trivial solution (a non-zero x such that Ax= 0) if and only if the kernel of A is non-trivial because the kernel of A is defined as the set such solutions. One is non-trivial if and only if the other is because they are, in fact, the same thing!

    Matrix A has inverse if an only if it's determinant is non-0. If A has an inverse then we can multiply both sides of Ax= 0 by it to get [itex]A^{-1}Ax= A^{-1}0[/itex] of [itex]x= 0[/itex] so the kerne is trivial, consisting only of 0.
     
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