Question

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:

• A. \(\frac{1}{4}\)
• B. \(\frac{3}{8}\)
• C. \(\frac{5}{8}\)
• D. \(\frac{3}{4}\)

Hint:

For variance problems involving matrices, ensure all possible determinant values are calculated correctly with corresponding probabilities.

Answer 1

The Correct Option is B

Step 1: Identifying possible determinant values. The determinant is calculated as: \[ |A| = a_{11}a_{22} - a_{12}a_{21} \] Using all possible combinations of 0 and 1, the possible determinant values are: \(\{-1, 0, 1\}\)

Step 2: Probability distribution. - Probability for \(X = -1\) = \(\frac{3}{16}\)
- Probability for \(X = 0\) = \(\frac{10}{16} = \frac{5}{8}\)
- Probability for \(X = 1\) = \(\frac{3}{16}\)

Step 3: Calculating Variance. \[ \text{Variance} = E(X^2) - (E(X))^2 \] \[ = \frac{3}{16}(-1)^2 + \frac{5}{8}(0)^2 + \frac{3}{16}(1)^2 - (0)^2 \] \[ = \frac{3}{16} + 0 + \frac{3}{16} = \frac{6}{16} = \frac{3}{8} \]