URGENT: adjoints, matrices

**eiktmywib** · October 18th 2008, 04:39 PM

If A is 3x3 and detA = 1/3
What is det(adjA)^-1?

**o_O** · October 18th 2008, 05:32 PM

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.

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