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Inverse Matrix Question

  1. Dec 1, 2008 #1
    1. The problem statement, all variables and given/known data

    Let A be an invertible 3x3 matrix. Suppose it is known that:
    A =
    [u v w
    3 3 -2
    x y z]
    and that adj(A) =
    [a 3 b
    -1 1 2
    c -2 d]
    Find det(A)

    (answer without any unknown variables)

    2. Relevant equations

    3. The attempt at a solution

    I found A^(-1) to be equal to
    (1/det(A)) * adj(A)

    So, then re-arranging the formula;
    I*det(A) = A*adj(A)
    So then det(A) =
    [u v w
    3 3 -2
    x y z]
    [a 3 b
    -1 1 2
    c -2 d]

    I know the solution to this problem is det(A) = 16. Therefore it must be that
    [3 3 -2] * [3 1 2] = det(A)

    But, what I am confused about is:
    Why is the det equal only to the position (2,2) of the matrix A*adj(A)? As the solution is taken as the position a_(2,2)...

  2. jcsd
  3. Dec 1, 2008 #2


    User Avatar
    Science Advisor
    Homework Helper

    If I*det(A)=A*adj(A) then you can evaluate any diagonal element to figure out det(A). a_(2,2) just happens to be the one you can figure out that doesn't have any unknowns in it. No big mysteries here.
  4. Dec 1, 2008 #3
    Thanks again Dick for clearing this up for me!
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