<p>How many different diagonal matrices of order <em>n</em> can be formed which are involuntary?</p>
Step-by-Step Solution
Key Concept: An involuntary matrix satisfies A² = I, so for diagonal matrices with entries d₁, d₂, ..., dₙ on the diagonal, each dᵢ² = 1, meaning each diagonal entry must be either +1 or -1. This gives 2 independent choices per diagonal position.
<p><strong>Step 1:</strong> Recall that a matrix A is involuntary if A² = I (A is its own inverse).</p><p><strong>Step 2:</strong> For a diagonal matrix D = diag(d₁, d₂, ..., dₙ), we have D² = diag(d₁², d₂², ..., dₙ²).</p><p><strong>Step 3:</strong> For D² = I, we need diag(d₁², d₂², ..., dₙ²) = I, which means d₁² = d₂² = ... = dₙ² = 1.</p><p><strong>Step 4:</strong> Each dᵢ must satisfy dᵢ² = 1, so dᵢ ∈ {+1, -1}.</p><p><strong>Step 5:</strong> Each of the n diagonal positions has exactly 2 independent choices (either +1 or -1).</p><p><strong>Step 6:</strong> By the multiplication principle, total number of involuntary diagonal matrices = 2 × 2 × 2 × ... × 2 (n times).</p><p><strong>∴ Answer: 2ⁿ</strong></p>
Correct Answer: 2^n