Probability
Classical probability
Grade 12
Question:
<p>Matrices of order \(2 \times 2\) are formed by using the elements of the set \(A = \{-2, -1, 0, 1, 2\}\), then probability that matrix is either symmetric or skew-symmetric, is greater than:</p>
<p>(a) \(\dfrac{1}{10}\)</p>
<p>(b) \(\dfrac{2}{10}\)</p>
<p>(c) \(\dfrac{3}{10}\)</p>
<p>(d) \(\dfrac{4}{10}\)</p>
Step-by-Step Solution
Key Concept: A matrix is symmetric if A^T = A and skew-symmetric if A^T = -A. For 2×2 matrices, symmetry requires a₁₂ = a₂₁, while skew-symmetry requires a₁₂ = -a₂₁ AND diagonal elements = 0. These conditions are mutually exclusive except for the zero matrix.
<p><strong>Step 1: Count total matrices</strong></p><p>Total 2×2 matrices with elements from {-2, -1, 0, 1, 2}: 5⁴ = 625</p><p><strong>Step 2: Count symmetric matrices</strong></p><p>Form: [[a, b], [b, c]] where a, b, c ∈ A</p><p>Choices: a (5 ways) × b (5 ways) × c (5 ways) = 125 symmetric matrices</p><p><strong>Step 3: Count skew-symmetric matrices</strong></p><p>Form: [[0, b], [-b, 0]] where b ∈ A (diagonal must be 0)</p><p>Choices: b (5 ways) = 5 skew-symmetric matrices</p><p><strong>Step 4: Check overlap</strong></p><p>For a matrix to be both symmetric and skew-symmetric: b = -b (so b = 0) and a = 0, c = 0</p><p>Only the zero matrix satisfies both conditions (counted in both sets)</p><p><strong>Step 5: Apply inclusion-exclusion</strong></p><p>Favorable outcomes = 125 + 5 - 1 = 129</p><p><strong>Step 6: Calculate probability</strong></p><p>P = 129/625 = 0.2064</p><p>∴ The probability is greater than 0.20 (or 20/100), Answer: <strong>B</strong></p>
Correct Answer: B