Let A=[aij] ​ be a 2x2 matrix such that aij ∈ {0, 1} for all i and j .

Question

Let \(\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right]\) be a \(2 \times 2\) matrix such that \(\mathrm{a}_{\mathrm{ij}} \in\{0,1\}\) 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:

(1) \(\frac{1}{4}\)

(2) \(\frac{3}{8}\)

(3) \(\frac{5}{8}\)

(4) \(\frac{3}{4}\)

Answer 1

Correct option is (2) \(\frac{3}{8}\)  

\(|A|=\left|\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right|\)  

\(=a_{11} a_{22}-a_{21} a_{12}\) 

\(=\{-1,0,1\}\)

\(\therefore \operatorname{var}(\mathrm{x})=\sum \mathrm{P}_{\mathrm{i}} \mathrm{X}_{\mathrm{i}}{ }^{2}-\left(\sum \mathrm{P}_{\mathrm{i}} \mathrm{X}_{\mathrm{i}}\right)^{2}\)

\(=\frac{3}{8}-0=\frac{3}{8}\)