Linear Algebra Proof: Skew Symmetric Matrix and Odd Number Determinant

[_Bd_]

Homework Statement

If A is a skew symetric matrix (such that A^T = -A)
and A is an nxn matrix with n being an odd number proove that det(A) = 0

The Attempt at a Solution

all I can think of is
det(A) = det(A^T)
letting c = -1 det(cA) = c^ndet(A)

but I can't get anymore connections to proove this. . .I tried doing some random example of a 3x3 skew symetric matrix but I didnt get a det=0. . .so I have no clue how to go about this problem!

 

[Dick]

If you took the determinant of a 3x3 skew symmetric matrix and didn't get zero, then you made a mistake. You basically just wrote down the proof. det(A)=det(A^T)=det(-A)=(-1)^n*det(A). Doesn't that show det(A)=0?

 

[_Bd_]

If you took the determinant of a 3x3 skew symmetric matrix and didn't get zero, then you made a mistake. You basically just wrote down the proof. det(A)=det(A^T)=det(-A)=(-1)^n*det(A). Doesn't that show det(A)=0?

i don't know. . .I don't see it

det(A)=det(A^T) which means just that if the determinant of A is 4 the determinant of A^T is 4

and the det(-A) =(-1)^n * det(A) which just doesn't mean anything . . .or I don't see it meaning anything? cause i don't know I am thinking about it as say the det of some matrix is 4 then the determinant of the negative of that matrix is 4 * (-1)^n . . .which I still don't see it as a zero. . .?
or maybe my train of thought is wrong? i don't know maybe I did make a mistake in my calculator

 

[Dick]

i don't know. . .I don't see it

det(A)=det(A^T) which means just that if the determinant of A is 4 the determinant of A^T is 4

and the det(-A) =(-1)^n * det(A) which just doesn't mean anything . . .or I don't see it meaning anything? cause i don't know I am thinking about it as say the det of some matrix is 4 then the determinant of the negative of that matrix is 4 * (-1)^n . . .which I still don't see it as a zero. . .?
or maybe my train of thought is wrong? i don't know maybe I did make a mistake in my calculator

The point is that (-1)^n=(-1) if n is odd. Since you then have det(A)=(-det(A)), what's the only possible value for det(A)?

 

[_Bd_]

oooh! now I get it... thanks!