Let $A=\left[a_{i j}\right]$ be a $2 \times 2$ matrix such that $a_{i j} \in\{0,1\}$ 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 :

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YESPYQ · Accepted Answer

Correct answer: b) $\frac{3}{8}$. $\left|A\right|=\left|\begin{matrix}{a}_{11} & {a}_{12} \ {a}_{21} & {a}_{22}\end{matrix}\right|$ $={a}_{11}{a}_{22}-{a}_{21}{a}_{12}\ \text{possible values for X}={-1,0,1}$ $\begin{matrix}\mathrm{x} & {\mathrm{P}}_{\mathrm{i}} & {\mathrm{P}}_{\mathrm{i}}{\mathrm{X}}_{\mathrm{i}} & {\mathrm{P}}_{\mathrm{i}}{{\mathrm{X}}_{\mathrm{i}}}^{2} \ -1 & \frac{3}{16} & -\frac{3}{16} & \frac{3}{16} \ 0 & \frac{10}{16} & 0 & 0 \ 1 & \frac{3}{16} & \frac{3}{16} & \frac{3}{16} \ & & \sum {\mathrm{P}}_{\mathrm{i}}{X}_{\mathrm{i}}=0 & \sum {\mathrm{P}}_{\mathrm{i}}{\mathrm{X}}_{\mathrm{i}}^{2}=\frac{3}{8}\end{matrix}$ $∴var(\mathrm{x})=\sum {\mathrm{P}}_{\mathrm{i}}{\mathrm{X}}_{\mathrm{i}}^{2}-{\left(\sum {\mathrm{P}}_{\mathrm{i}}{\mathrm{X}}_{\mathrm{i}}\right)}^{2}\ =\frac{3}{8}-0=\frac{3}{8}$

Let \(A=\left[a_{i j}\right]\) be a \(2 \times 2\) matrix such that \(a_{i j} \in\{0,1\}\) for all \(i\) and \(j\). Let …

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