# Math Help - Linear algebra

1. ## Linear algebra

Let $A \in M_n(k)$ i
1 A is Invertible. Calculate Det(adj A)
2 If A is antisimetric and n is odd, Prove that Det = 0

2. Originally Posted by kezman
Let $A \in M_n(k)$ invertible matrix.
If $A$ is an invertible matrix. Then, $A^{-1} = \frac{1}{\det (A)}\mbox{adj}(A)$. Thus, $\det (A)A^{-1} = \mbox{adj}(A)$. Thus, $\det (\det(A) A^{-1} ) = \det (\mbox{adj}(A))$. Thus, $[\det(A)]^k \cdot \frac{1}{\det (A)} = \det (\mbox{adj}(A))$.
3. 2) If $A$ is antisimetric then $A=-A^T$
Then $\det A=\det(-A^T)=(-1)^n\det A^T=-\det A\Rightarrow \det A=0$