<p>Find the value of the determinant \(\begin{vmatrix} 1 & 1 & 1 & 1 \\ 1 & 2 & 3 & 4 \\ 1 & 3 & 6 & 10 \\ 1 & 4 & 10 & 20 \end{vmatrix}\)</p>
Step-by-Step Solution
Key Concept: Recognize that rows contain triangular numbers and cumulative sums; use row operations (subtract successive rows) to reveal a pattern that simplifies to a product of differences.
<p><strong>Step 1: Identify the structure.</strong> The matrix contains:</p><ul><li>Row 1: all 1's</li><li>Row 2: 1, 2, 3, 4 (natural numbers)</li><li>Row 3: 1, 3, 6, 10 (triangular numbers)</li><li>Row 4: 1, 4, 10, 20 (tetrahedral/tetrahedral sums)</li></ul><p><strong>Step 2: Apply row operations.</strong></p><p>R₂ → R₂ - R₁: [0, 1, 2, 3]</p><p>R₃ → R₃ - R₂: [0, 1, 3, 6]</p><p>R₄ → R₄ - R₃: [0, 1, 4, 10]</p><p><strong>Step 3: Expand along column 1.</strong> Since column 1 now has three zeros:</p><p>det = 1 · ∣∣ 1 2 3 ∣∣</p><p> ∣∣ 1 3 6 ∣∣</p><p> ∣∣ 1 4 10 ∣∣</p><p><strong>Step 4: Repeat row operations on the 3×3 matrix.</strong></p><p>R₂ → R₂ - R₁: [0, 1, 3]</p><p>R₃ → R₃ - R₂: [0, 1, 4]</p><p>Expand along column 1:</p><p>det = 1 · ∣∣ 1 3 ∣∣ = 1 · (1·4 - 3·1) = 1 · 1 = 1</p><p> ∣∣ 1 4 ∣∣</p><p><strong>∴ Answer: 1</strong></p>
Correct Answer: 1