Determinants

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September 14th 2008, 05:54 PM
mivanova

Determinants

Hi,
Can you help me with this problem, please
Suppose that A is nxn with n odd and that AT=-A. Show that
(i)If 2≠0 in K then det(A)=0
(ii) If 2=0 but we assume that each diagonal entry of A equals 0, then det(A)=0
Thank you!
September 14th 2008, 05:58 PM
ThePerfectHacker

Quote:

Originally Posted by mivanova

Hi,
Can you help me with this problem, please
Suppose that A is nxn with n odd and that AT=-A. Show that
(i)If 2≠0 in K then det(A)=0
(ii) If 2=0 but we assume that each diagonal entry of A equals 0, then det(A)=0
Thank you!

$\det (A) = \det (A^T) =\det (-A) = (-1)^n \det (A) = -\det (A)$
Thus, $2\det (A) = 0$ .

Now if $\text{char}(K)\not = 2$ then it must mean $\det (A) = 0$ .

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