Let $\displaystyle A \in M_n(k) $ i
1 A is Invertible. Calculate Det(adj A)
2 If A is antisimetric and n is odd, Prove that Det = 0
If $\displaystyle A$ is an invertible matrix. Then, $\displaystyle A^{-1} = \frac{1}{\det (A)}\mbox{adj}(A)$. Thus, $\displaystyle \det (A)A^{-1} = \mbox{adj}(A)$. Thus, $\displaystyle \det (\det(A) A^{-1} ) = \det (\mbox{adj}(A))$. Thus, $\displaystyle [\det(A)]^k \cdot \frac{1}{\det (A)} = \det (\mbox{adj}(A))$.