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Properties of Determinants

  1. Nov 15, 2013 #1
    1. The problem statement, all variables and given/known data

    If A is a 2x2 matrix, then det (2A * adj(A)^-1) = ?

    2. Relevant equations

    Adj(A)A = det(A)I

    3. The attempt at a solution

    First, I separated them so it became det(2A) * det (1/ adj(A))
    Then taking the 2 out, and it becomes 2^2, so 4 det(A) * det(1/ adj(A))
    adj(A) = det(A) * A^-1, rearranged from the equation above.
    So: 4 det(A) * det (A/det(A)), and I get stuck at around here, because I end with
    4 det(A) ^2 * det (1/det(A)), however I don't know what the determinant of the determinant of A is. Could someone clarify this for me? thank you in advance.
     
  2. jcsd
  3. Nov 15, 2013 #2

    Dick

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    If it's a 2x2 matrix then det(I/det(A))=1/det(A)^2, yes?
     
  4. Nov 15, 2013 #3
    The determinant of A is just a number, like 2 was. :smile:
     
  5. Nov 15, 2013 #4
    Ohhhh! This was actually really helpful to another question I had.... so if it was a 3x3 matrix, the det(1+2detA) where detA = 2 would become det(5), and since the matrix is 3x3, det5 becomes 5^3 = 125?

    Oh one more thing, it would be the exact same as in (1+2detA)^3
     
    Last edited: Nov 15, 2013
  6. Nov 15, 2013 #5
    Hmm, not precisely. 1 + 2det(A) is a number, not a matrix, so its determinant would have to be itself, if we identified it as a 1x1 matrix. If you had to compute det((1 + 2det(A))*A) = det(5*A) where A is a 3x3 matrix, then you would get (5^3)*det(A), through the usual theorem on computing the determinant of a matrix multiplied by a number.
     
  7. Nov 16, 2013 #6
    I see, oh I wrote it wrong, I meant the question was det (((1 +2det(A))* I), would it be considered a coefficient of I in this case and become 5^3 det(I)?
     
  8. Nov 16, 2013 #7

    Dick

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    Yes, that's it. And I'm sure you know what det(I) is.
     
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