Expected determinant of a random symmetric matrix

Question

The three distinct entries of a $2 \times 2$ symmetric matrix are drawn from the uniform distribution over $[-60, 60]$. What is the expected determinant of the matrix?

I assume it is $0$ but I am not good at proving it efficiently. Thanks.

Answer 1

Let the matrix be in the form of

$$\begin{bmatrix}a & b \\ b & d \end{bmatrix}$$

The determinant is $ad-b^2$.

Assuming the $3$ entries to be independent. \begin{align}\mathbb{E}[ad-b^2]&=\mathbb{E}[ad]-\mathbb{E}[b^2]\\&=\mathbb{E}[a]\mathbb{E}[d]-\mathbb{E}[b^2] \\ &=-(\mathbb E[b^2]-\mathbb{E}[b]^2)\\ &=-Var(b)\end{align}