Expected Value of a Determinant

Suppose that I construct an $n \times n$ matrix $A$ such that each entry of $A$ is a random integer in the range $[1, \, n]$. I'd like to calculate the expected value of $\det(A)$.

My conjecture is that the answer is zero, though I could very well be incorrect. Running some numerical experiments with different values for $n$ and a large number of trials, it seems that $\mathbb{E}[\det(A)]$ is normally in the range $[0.25, \, 0.7]$, so I'm starting to lose faith in my intuition that it is zero.

Could anyone lend some advice on how to approach this problem and what strategies I may want to consider applying?

Rebecca's answer is nice, but here's another solution that might be simpler for some people: Let $f$ be a function that swaps the first two rows. Notice that $f(A)$ and $A$ have the same distribution, and thus $$\mathbb{E}[\det A] = \mathbb{E}[\det f(A)] = \mathbb{E}[-\det A] = - \mathbb{E}[\det A].$$

For $n \geq 2$, we partition the matrices into "orbits" formed by swapping the first two rows:

• Orbits of size $1$ have two identical rows, so $\det=0$.

• Orbits of size $2$ have matrices of determinants of equal magnitude but opposite sign.

Hence $\sum_A \det(A)=0$ and so $\mathbb{E}(\det(A))=\tfrac{1}{n^{n^2}}\sum_A \det(A)=0$.