Let A = [a_ij] be a 2 2 matrix such that a_ij 0, 1 for all i and j.

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

$$\begin{aligned}
& |A|=\left|\[\begin{array}{ll} a_{11} & a_{12} \ a_{21} & a_{22} \end{array}\end{aligned}$$\right| \
\& =a\_{11} a\_{22}-a\_{21} a\_{12} \
\& =\{-1,0,1\}
\]

$$\[\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} \  \ & \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}$$\]

$$\begin{aligned}
& \therefore \operatorname{var}(\mathrm{x})=\sum \mathrm{P}_{\mathrm{i}} \mathrm{X}_{\mathrm{i}}^2-\left(\sum \mathrm{P}_{\mathrm{i}} X_{\mathrm{i}}\right)^2 \
& =\frac{3}{8}-0=\frac{3}{8}
\end{aligned}$$

Solutions · 1