Expected determinant of A minus its transpose

Question

Fill a $2\times2$ matrix $A$ with independent entries, each equal to $0$ or $1$ with probability $\tfrac12$ . What is $E[\det(A-A^{T})]$ ?

Short answer

$1/2$ . $A-A^T$ is skew-symmetric with zero diagonal and off-diagonal entries $\pm(A_{12}-A_{21})$ , so its determinant is $(A_{12}-A_{21})^2$ . That equals $1$ when the two off-diagonal entries differ, which happens with probability $\tfrac12$ .

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Lessons behind this question

Expectation, Mean, and Variance
Conditional Expectation and Variance Revisited
The Determinant: Defining Properties

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