Determinant after antisymmetrizing

EasyProbability Linear Algebra~4m

Let $A$ be a random $2 \times 2$ matrix whose four entries are independent $\text{Bernoulli}(\tfrac12)$ (each $0$ or $1$ ). Compute $\mathbb{E}\big[\det(A - A^{\top})\big]$ .

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$A-A^\top$ is antisymmetric: zeros on the diagonal, $\pm(A_{12}-A_{21})$ off it.
Its determinant is $(A_{12}-A_{21})^2$ , which is $1$ iff the two bits differ.

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Determinant after antisymmetrizing

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