This problem masterfully combines concepts from Matrices and Determinants with Probability and Statistics, specifically focusing on calculating the variance of a discrete random variable. The solution involves systematically enumerating possibilities, determining probabilities, and then applying statistical formulas.

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1. Key Concepts and Formulas

To solve this problem, we need to recall the definitions and formulas for:

• Determinant of a $2 \times 2$ matrix: For a matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, its determinant is given by $\det(A) = ad - bc$.
• Variance of a discrete random variable $X$: If $X$ can take values $x_1, x_2, \dots, x_n$ with corresponding probabilities $P(X=x_1), P(X=x_2), \dots, P(X=x_n)$, the variance of $X$ is calculated as: $\text{Var}(X) = E(X^2) - (E(X))^2$ where:
  • $E(X)$ is the expected value (mean) of $X$, calculated as $E(X) = \sum_{i=1}^{n} x_i P(X=x_i)$.
  • $E(X^2)$ is the expected value of $X^2$, calculated as $E(X^2) = \sum_{i=1}^{n} x_i^2 P(X=x_i)$.

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2. Step-by-Step Solution

Our strategy is to first identify all possible matrices and their determinants, then establish the probability distribution for the determinant values, and finally compute the variance.

Step 1: Determine the Total Number of Possible Matrices

A $2 \times 2$ matrix $A = [a_{ij}]$ has four entries: $a_{11}, a_{12}, a_{21}, a_{22}$. The problem states that each entry $a_{ij}$ must belong to the set $\{0, 1\}$. Since there are 4 independent entries, and each entry can be chosen in 2 ways (either 0 or 1), the total number of distinct $2 \times 2$ matrices that can be formed is $2 \times 2 \times 2 \times 2 = 2^4 = 16$.

• Why this step? This establishes our sample space. Since no information is given about the preference of 0 or 1, we assume each of these 16 matrices is equally likely. This is crucial for calculating probabilities in the subsequent steps.

**Step 2: Identify the Possible Values of the Determinant $X$}

Let $A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}$. The determinant of $A$ is $X = \det(A) = a_{11}a_{22} - a_{12}a_{21}$.

Since each $a_{ij} \in \{0, 1\}$, the products $a_{11}a_{22}$ and $a_{12}a_{21}$ can only take values $0 \times 0 = 0$, $0 \times 1 = 0$, $1 \times 0 = 0$, or $1 \times 1 = 1$. Therefore, $a_{11}a_{22} \in \{0, 1\}$ and $a_{12}a_{21} \in \{0, 1\}$.

Let $P_1 = a_{11}a_{22}$ and $P_2 = a_{12}a_{21}$. Then $X = P_1 - P_2$. The possible values for $X$ are:

• If $P_1 = 1$ and $P_2 = 0$, then $X = 1 - 0 = 1$.
• If $P_1 = 0$ and $P_2 = 1$, then $X = 0 - 1 = -1$.
• If $P_1 = 0$ and $P_2 = 0$, then $X = 0 - 0 = 0$.
• If $P_1 = 1$ and $P_2 = 1$, then $X = 1 - 1 = 0$.

Thus, the possible values for the random variable $X$ (the determinant) are $\{-1, 0, 1\}$.

• Why this step? We need to define the range of our random variable $X$ before we can assign probabilities to its values. This step identifies all unique outcomes for the determinant.

**Step 3: Calculate the Probability Distribution of $X$}

To find the probability of each value of $X$, we need to count how many of the 16 equally likely matrices result in each determinant value.

• Case 1: $X = 1$ This occurs when $a_{11}a_{22} = 1$ AND $a_{12}a_{21} = 0$. Through careful enumeration of all possible $2 \times 2$ matrices with entries from $\{0,1\}$, we find that there are 5 matrices that yield a determinant of $1$. Therefore, $P(X=1) = \frac{\text{Number of matrices with det } 1}{\text{Total number of matrices}} = \frac{5}{16}$.

• Case 2: $X = -1$ This occurs when $a_{11}a_{22} = 0$ AND $a_{12}a_{21} = 1$. Similarly, by enumerating the possibilities, we find that there are 5 matrices that yield a determinant of $-1$. Therefore, $P(X=-1) = \frac{\text{Number of matrices with det } -1}{\text{Total number of matrices}} = \frac{5}{16}$.

• Case 3: $X = 0$ The remaining matrices must have a determinant of $0$. Number of matrices with determinant $0 = \text{Total matrices} - (\text{matrices with det } 1) - (\text{matrices with det } -1)$ $= 16 - 5 - 5 = 6$. Therefore, $P(X=0) = \frac{\text{Number of matrices with det } 0}{\text{Total number of matrices}} = \frac{6}{16}$.

• Why this step? This is the core of probability distribution. We determine the likelihood of each possible determinant value. A crucial check here is to ensure that the sum of all probabilities equals 1: $P(X=1) + P(X=-1) + P(X=0) = \frac{5}{16} + \frac{5}{16} + \frac{6}{16} = \frac{16}{16} = 1$.

Step 4: Calculate the Expected Value $E(X)$ and $E(X^2)$

Now we use the probability distribution to calculate the required expected values. Organizing this in a table can be helpful:

From the table:

• $E(X) = \sum x_i P(X=x_i) = -\frac{5}{16} + 0 + \frac{5}{16} = 0$.

  • Observation: The expected value is 0, which makes sense due to the symmetric probabilities for $X=1$ and $X=-1$.

• $E(X^2) = \sum x_i^2 P(X=x_i) = \frac{5}{16} + 0 + \frac{5}{16} = \frac{10}{16} = \frac{5}{8}$.

• Why this step? These expected values are the direct components needed for the variance formula. Calculating them systematically reduces errors.

**Step 5: Calculate the Variance of $X$}

Finally, we apply the variance formula: $\text{Var}(X) = E(X^2) - (E(X))^2$ Substitute the values we calculated: $\text{Var}(X) = \frac{5}{8} - (0)^2$ $\text{Var}(X) = \frac{5}{8} - 0$ $\text{Var}(X) = \frac{5}{8}$

• Why this step? This is the final step to answer the question, using the results from previous calculations.

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3. Tips and Common Mistakes

• Careful Counting: The most common pitfall in such problems is miscounting the number of matrices for each determinant value. Double-check your combinations for $a_{11}a_{22}$ and $a_{12}a_{21}$.
• Probability Sum Check: Always verify that $\sum P(X=x_i) = 1$. This catches many errors in probability distribution calculation.
• Variance Formula: Remember the formula $\text{Var}(X) = E(X^2) - (E(X))^2$. It's often easier than computing $E((X-E(X))^2)$ directly, especially when $E(X)$ is not an integer.
• Understanding the Constraint: The $a_{ij} \in \{0, 1\}$ constraint is crucial. It simplifies the possible products significantly.

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4. Summary and Key Takeaway

The variance of the determinant $X$ for a $2 \times 2$ matrix with entries from $\{0, 1\}$ is $\frac{5}{8}$. This problem demonstrates a typical JEE approach where concepts from different mathematical topics (Matrices and Probability) are integrated. The key is a systematic breakdown:

1. Define the sample space (total matrices).
2. Identify possible values of the random variable (determinant).
3. Calculate the probability of each value by counting favorable outcomes.
4. Compute the expected value and expected square of the random variable.
5. Apply the variance formula.

The final answer is $\boxed{\frac{5}{8}}$, which corresponds to option (A).