If A is an n x n matrix, where n is an odd integer, which satisfies A^T=-A (ie its skew-symmetric). Show that A has no inverse...........i can understand why but i cant show it mathematically

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- Apr 20th 2008, 02:36 AMflawlessLinear algebra. Showing n x n matrix has no inverse
If A is an n x n matrix, where n is an odd integer, which satisfies A^T=-A (ie its skew-symmetric). Show that A has no inverse...........i can understand why but i cant show it mathematically

- Apr 20th 2008, 07:26 AMred_dog
Note that $\displaystyle \det(^tA)=\det(A)$ and $\displaystyle \det(aA)=a^n\det(A)$.

Now, applying the determinant in the equality we have

$\displaystyle \det(^tA)=(-1)^n\det(A)\Rightarrow \det(A)=-\det(A)\Rightarrow 2\det(A)=0\Rightarrow\det(A)=0$