Let A = [a_{ij}] be a 2 imes 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:

Question

Let A = [a_{ij}] be a 2 	imes 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:

Accepted Answer

Answer: D. \frac{3}{8} \begin{aligned} & |A|=\left|\begin{array}{ll} a_{11} & a_{12} \ a_{21} & a_{22} \end{array}ight| \ & =a_{11} a_{22}-a_{21} a_{12} \ & =\{-1,0,1\} \end{aligned} \begin{array}{c|c|c|c} \mathrm{x} & \mathrm{P}_{\mathrm{i}} & \mathrm{P}_{\mathrm{i}} \mathrm{X}_{\mathrm{i}} & \mathrm{P}_1 \mathrm{X}_{\mathrm{i}}{ }^2 \ -1 & \frac{3}{16} & -\frac{3}{16} & \frac{3}{16} \ 0 & \frac{10}{16} & 0 & 0 \ 1 & \frac{3}{16} & \frac{3}{16} & \frac{3}{16} \ \hline & & \sum \mathrm{P}_{\mathrm{i}} \mathrm{X}_{\mathrm{i}}=0 & \sum \mathrm{P}_{\mathrm{i}} \mathrm{X}_{\mathrm{i}}{ }^2=\frac{3}{8} \end{array} \begin{aligned} & 	herefore \operatorname{var}(\mathrm{x})=\sum \mathrm{P}_{\mathrm{i}} \mathrm{X}_{\mathrm{i}}^2-\left(\sum \mathrm{P}_{\mathrm{i}} X_{\mathrm{i}}ight)^2 \ & =\frac{3}{8}-0=\frac{3}{8} \end{aligned}

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:

Answer

Explanation

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