Variance of a Certain Random Matrix 
Let $A$ be a random $n \times n$ matrix, whose entries $X_{ij}$ are independent and $P(X_{ij}=1)=P(X_{ij}=-1)=1/2$. Compute $\text{Var}(\text{det}(A))$.

I'm not really sure how to even proceed here. I think the question may imply that each $X_{ij}$ is either $1$ or $-1$ almost surely. So perhaps that helps things out.
I would really appreciate any help with this question.
 A: By definition of the determinant:
$$\det(A)=\sum_{\sigma\in S_n}\varepsilon(\sigma)X_{1,\sigma(1)}\cdots X_{n,\sigma(n)}$$
The random variables $Y_\sigma=\varepsilon(\sigma)X_{1,\sigma(1)}\cdots X_{n,\sigma(n)}$ are not independent, but we have:
$$V(\det(A))=\sum_{\sigma\in S_n}V(Y_{\sigma})+\sum_{\sigma\neq\sigma'}Cov(Y_\sigma,Y_{\sigma'})$$
Now we can show that 


*

*$V(Y_\sigma)=1$ for any $\sigma\in S_n$

*$Cov(Y_\sigma,Y_{\sigma'})=0$ for $\sigma\neq\sigma'$


This will prove that $$V(\det(A))=n!$$
First, $V(Y_\sigma)=V(X_{1,\sigma(1)}\cdots X_{n,\sigma(n)})$ since $|\varepsilon(\sigma)|=1$. I'll still denote this by $Y_{\sigma}$.
By independance of the $X_{ij}$, we get that
$$V(X_{1,\sigma(1)}\cdots X_{n,\sigma(n)})=\prod_{i=1}^nE(X_{i,\sigma(i)}^2)-\left(\prod_{i=1}^nE(X_{i,\sigma(i)})\right)^2$$
$E(X_{i,\sigma(i)}^2)=1$ since $X_{i,\sigma(i)}^2=1$ a.s., and $E(X_{i,\sigma(i)})=0$. So $V(Y_\sigma)=1$.
Now $Cov(Y_\sigma,Y_{\sigma'})=E(Y_\sigma Y_{\sigma'})-E(Y_\sigma)E(Y_{\sigma'})$. 
By independence, $E(Y_\sigma)=E(Y_{\sigma'})=0$. Then
$$E(Y_\sigma Y_{\sigma'})=E\left(X_{1,\sigma(1)}\cdots X_{n,\sigma(n)}X_{1,\sigma'(1)}\cdots X_{n,\sigma'(n)}\right)$$
As $\sigma\neq\sigma'$, there exists $i\in\{1,\dots,n\}$ such that $\sigma(i)\neq\sigma'(i)$. Then $X_{i,\sigma(i)}$ is independent from all the other variables in the product so you can factor its expected value which is $0$. We just proved that $Cov(Y_\sigma,Y_{\sigma'})=0$.
