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Math Help - Adjoints and Determinants Help

  1. #1
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    Adjoints and Determinants Help

    Adjoints and Determinants Question?

    1) Let A be a 3x3 matrix with determinant -12.

    a) What is det(adj(A^T))?
    b) What is det(adj(A^-1))?
    c) What is det(adj(3A))?

    2) Let A be a 2x2 matrix such that adj(A) =

    [8, -1]
    [-5, -4]

    What is det(A)?
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  2. #2
    o_O
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    It can be shown that: \text{det} \left(\text{adj}(A)\right) = \left[\text{det} (A) \right]^{n-1}

    So for #1, just use this directly and simplify using your properties of determinants:
    (a): \text{det} \left(\text{adj}(A^T)\right) = \left[\det(A^T)\right]^{n-1} = \cdots

    (b): \text{det} \left(\text{adj}(A^{-1})\right) = \left[\det(A^{-1})\right]^{n-1}= \cdots

    (c): \text{det} \left(\text{adj}(3A)\right) = \left[\det(3A)\right]^{n-1}= \cdots

    For #2, modifying the property I gave you earlier, we get: \det (A) = \sqrt[n-1]{\det \left(\text{adj} (A)\right)}

    So find the determinant of your adjoint matrix and then take its (n-1)-th root.
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