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Math Help - URGENT: adjoints, matrices

  1. #1
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    URGENT: adjoints, matrices

    If A is 3x3 and detA = 1/3
    What is det(adjA)^-1?
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  2. #2
    o_O
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    Note that If A is invertible, then: A^{-1} = \frac{\text{adj}(A)}{\text{det}(A)}

    From here, you can derive an expression for \text{det}(\text{adj}(A)):

    \text{adj}(A) = \text{det}(A) \cdot A^{-1} ...............................multiplied both sides by \text{det}(A)

    \text{det} \left(\text{adj}(A)\right) = \text{det} \left( \text{det} (A) \cdot A^{-1}\right) ...............took the determinant of both sides

    \text{det} \left(\text{adj}(A)\right) = \left[\text{det}(A) \right]^{3} \cdot \text{det} (A^{-1}) .............since \text{det}(cA) = c^{n} \text{det}(A)

    \text{det} \left(\text{adj}(A)\right) =  \left[\text{det}(A) \right]^{3} \cdot \frac{1}{\text{det}(A)}................since \text{det}(A^{-1}) = \frac{1}{\text{det}(A)}

    \text{det} \left(\text{adj}(A)\right) = \hdots etc.
    Last edited by o_O; October 18th 2008 at 06:26 PM.
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