Solving Odd n x n Anti-Symmetric Matrices | Using Theorem 3.5

[DWill]

Homework Statement

An n x n matrix A is anti-symmetric if it satisfies the equation A^t = -A.
Show that if n is odd and A is anti-symmetric, then det(A) = 0. (Hint: carefully
use Theorem 3.5 on page 187.)

Homework Equations

Theorem 3.5: If B is obtained from A by multiplying a row (column) of A by a real number c, then det(B) = c det(A).

A^t = inverse of A

The Attempt at a Solution

I found the general 3x3 antisymmetric matrix to look like this:

[
0 a_12 a_13
-a_12 0 a_23
-a_13 -a_23 0
]

To find the determinant I just used the method of left and right diagonals since the matrix is 3x3, and I find it to be 0. BUT I don't know how to show this for any n x n matrix with n being odd (so it can be 5x5, 7x7, etc). I don't know where to use the theorem given in the hint either. Please help! Thanks

 

[Dick]

Forget the specific dimension. You had better know that det(A)=det(A^t). Do you? If so, that tells you that det(A)=det(A^t)=det(-A). You can get -A from A by multiplying each row of A by -1, one at a time. Now what? Remember the number of row is odd.

 

[DWill]

I see, I get to this point:

det(-A) = (-1)^n * det(A)

I'm not sure where to go from there though. For an odd n det(-A) will be negative of det(A), how does this show det(A) = 0? thanks

 

[gabbagabbahey]

For an odd n det(-A) will be negative of det(A), how does this show det(A) = 0? thanks

If I told you that $x=-x$, would you be able to tell me what $x$ was?

P.S. You wouldn't happen to be a Utah Jazz fan would you ?

 

[Dick]

If you've got det(A^t)=det(A) and det(A^t)=det(-A) and det(-A)=(-1)^n*det(A), and you know (-1)^n=(-1), then you have det(A)=-det(A), right? Chain them all together. For what real number is x=(-x)? There's only one.

 

[Dick]

If I told you that $x=-x$, would you be able to tell me what $x$ was?

P.S. You wouldn't happen to be a Utah Jazz fan would you ?

Utah Jazz? I give up. I'm a sports ignoramus. Basketball player? Dwill? Am I getting close?

 

[gabbagabbahey]

Utah Jazz? I give up. I'm a sports ignoramus. Basketball player? Dwill? Am I getting close?

D-Will is the Nickname for Deron Williams; one of the best point guards (yes, that's basketball) in the NBA and a member of this years Gold medal winning US Olympic squad

 

[Mark44]

From Dick's hint you can say something about det(A^t), and from the given information, you can say something about (-1)^n.

 

[Dick]

I feel proud I knew it was basketball. Good thing this isn't the Sports Forum. I won't ask what a "point guard" is.

 

[DWill]

Haha wow I should've seen that one. :(

And gabba yes I am a Deron Williams fan, though not a Jazz fan. :) I don't hate them or anything, just neutral. I am actually rooting for Houston, and I can't wait to see the first game between them and the Jazz this year after being eliminated by them last few years. Utah also seems to have a thing against Ron Artest, so that will be fun to watch too. :)

Anyways, thanks a lot for the help, unfortunately I'll probably have many more questions to come since I'm really trying to catch up in my Linear Algebra class right now.

 

[Dick]

Watch less basketball. Do more linear algebra. Wouldn't that be more fun? Just kidding.

 

[gabbagabbahey]

I'm a Raptors fan myself, but You've got to respect a guy with Deron's talents...I was pretty big on Houston coming into the season, but after watching them get killed by the Lakers in the 4th quarter the other night, I think it's safe to say that they have an outside shot at best of getting to the finals.

 

[Dick]

I'm outta here.

 

[gabbagabbahey]

Watch less basketball. Do more linear algebra. Wouldn't that be more fun? Just kidding.

A healthy dose of both is my prescription (And playing basketball is even better)

 

[Dick]

Absolutement.