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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- Nov 16th 2008, 09:51 PMMy Little PonyAdjoints 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)? - Nov 16th 2008, 10:06 PMo_O
It can be shown that: $\displaystyle \text{det} \left(\text{adj}(A)\right) = \left[\text{det} (A) \right]^{n-1}$

So for #1, just use this directly and simplify using your properties of determinants:

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

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

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

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

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