# Linear algebra

• August 29th 2007, 07:47 PM
kezman
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
• August 29th 2007, 07:56 PM
ThePerfectHacker
Quote:

Originally Posted by kezman
Let $A \in M_n(k)$ invertible matrix.
1 Calculate Det(adj A)
2 If A is antisimetric and n is odd, Prove that Det = 0

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, $[\det(A)]^k \cdot \frac{1}{\det (A)} = \det (\mbox{adj}(A))$.
• August 30th 2007, 12:49 AM
red_dog
2) If $A$ is antisimetric then $A=-A^T$
Then $\det A=\det(-A^T)=(-1)^n\det A^T=-\det A\Rightarrow \det A=0$
• August 30th 2007, 03:39 PM
kezman
Thank you very much.