Let $A = [a_{ij}]$ be a $2 \times 2$ matrix such that $a_{ij} \in \{0, 1\}$ for all $i$ and $j$. Let the random variable $X$ denote the possible values of the determinant of the matrix $A$. Then, the variance of $X$ is:
Step-by-Step Solution
Key Concept: List all $2^4 = 16$ possible $2\times2$ matrices with 0/1 entries, compute each determinant, build the probability distribution of $X$, then find variance.
$P(X=0) = 10/16$, $P(X=1) = 3/16$ (e.g., $\begin{pmatrix}1&0\\0&1\end{pmatrix}$ type), $P(X=-1) = 3/16$. Mean $\mu = 0$. Var$(X) = E(X^2) = 1^2\cdot\frac{3}{16} + (-1)^2\cdot\frac{3}{16} = \frac{6}{16} = \frac{3}{8}$.
Correct Answer: $\frac{3}{8}$