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)?
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)?
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.