Why Does Tensor Contraction Yield Zero in This Calculation?

[teddd]

Homework Statement

Hello guys, hope you'll help me out with this!
I'm asked to calculate $$g^{\alpha\beta}g^{\sigma\rho}(g_{\alpha\sigma}g_{\beta\rho}-g_{\alpha\rho}g_{\beta\sigma})$$
where g is the metric tensor on a n-dimensional manifold but I can't get to the right result, i keep on getting zero! (i know that's wrong although i don't know the exact solution -sorry- but it should depend on the dimension of the manifold)

Homework Equations

none

The Attempt at a Solution

Well, i contracted firs with respect to $g^{\sigma\rho}$ and i end up with $$g^{\alpha\beta}(g_{alpha\beta}+g_{alpha\beta}-g_{alpha\beta}-g_{alpha\beta})$$which obviously vanish.
I must be missing something!

 

[hunt_mat]

Are you so sure it's zero? Let's look at the first factor:
$$g^{\alpha\beta}g^{\sigma\rho}g_{\alpha\sigma}g_{ \beta\rho}=g^{\alpha\beta}(g^{\sigma\rho}g_{\alpha\sigma})g_{\beta\rho}=g^{\alpha\beta}\delta^{\rho}_{\alpha}g_{\beta\rho}=g^{\rho\beta}g_{\beta\rho}=\delta^{\rho}_{\rho}=n$$
What is the other factor?

 

[teddd]

Thanks for the answer hunt_mat but... I keep on getting zero!

That's becaouse contracting the second factor like you did for the first i get
$$-g^{\alpha\beta} g^{\sigma\rho}g_{\alpha\rho}g_{\beta\sigma}=-g^{\alpha\beta} (g^{\sigma\rho}g_{\alpha\rho})g_{\beta\sigma}=-g^{\alpha\beta} \delta^{\sigma}_{\alpha}g_{\beta\sigma}=-g^{\sigma\beta}g_{\beta\sigma}=-n$$

which added to the first factor gives zero!
They seems equal to me, becaouse when i get the kronecker delta it doesent matter which index it has, it will be a mute index anyhow! (maybe the error is here?)where am i mistaking?

 

[hunt_mat]

I don't think that you're making a mistake at all, I think the answer really is zero.