If three random variables have the same variance, what will the co-variances look like? I'm curious to know, if three random variables have the same variance, what will the co-variances look like? Can somebody help me to figure it out?
 A: The covariances will be the common variance mutliplied by the correlations between the variables, which will be between $-1$ and $1$, though there are some restrictions on the possible values the set of three correlations might take (they can't all be $-1$ at the same time, for example).
So if the variables are $Y_1,Y_2$ and $Y_3$, the common variance is $\sigma^2$ and the three correlations are $\rho_{12},\rho_{13}$ and $\rho_{23}$, then the covariance $c_{ij} = \sigma^2 \rho_{ij}$ for each $i,j$ combination.
The set of possible values for the three correlations is such that either the correlation matrix or the covariance matrix (either implies the other given the variances are all positive) is positive semi-definite. A symmetric $n × n$ real matrix $A$ is positive semi-definite if $x^TAx \geq 0$ for every non-zero column vector $x$ of $n$ real numbers. 
(In your case, $n=3$ of course.)
So, for an example of a correlation matrix that cannot occur, consider the case where all the correlations $\rho_{12},\rho_{13}$ and $\rho_{23}$ are $-0.8$. Then if you take the vector $x$ above to be all $1$'s,  $x^TAx = -1.8$, so there's at least one $x$ for which $x^TAx$ isn't at least $0$, and so the matrix $A$ isn't positive semi-definite, and cannot be a correlation matrix.
In R:
> A
     [,1] [,2] [,3]
[1,]  1.0 -0.8 -0.8
[2,] -0.8  1.0 -0.8
[3,] -0.8 -0.8  1.0

> x
     [,1]
[1,]    1
[2,]    1
[3,]    1

> t(x) %*% A %*% x   #  x' A x
     [,1]
[1,] -1.8

So if we try to do anything that relies on it being positive semi-definite, it will fail:
> chol(A)
Error in chol.default(A) : 
  the leading minor of order 3 is not positive definite

[Why does that definition of what's a possible correlation matrix work? Consider that we define a new random variable, $Z = x^TY$. Then $\text{Var}(Z) = x^T\text{Var}(Y)x = \sigma^2 x^TAx$. Clearly, if $x^TAx$ could be negative, there'd be a random variable, $Z$ with a negative variance. So we need $x^TAx\geq 0$ for every possible $x$.]

edit: some more-or-less related questions:
The bound on a common correlation of three variables (the bounds on $\rho$ for the case $\rho_{12}=\rho_{13}=\rho_{23}=\rho$) is discussed in this question.
If you specify two correlations and look at the possible values of the third, there's some relevant discussion in this question.
