<p>Using properties of determinants, evaluate<br>\[\begin{vmatrix} 18 & 40 & 89 \\ 40 & 89 & 198 \\ 89 & 198 & 440 \end{vmatrix}\]</p>
Step-by-Step Solution
Key Concept: Recognize that each element follows a pattern where each entry is a sum of products from previous entries. Specifically, observe that 40 = 18 + 22, 89 = 40 + 49, 198 = 89 + 109, 440 = 198 + 242, revealing a recurrence structure. This allows you to express rows as linear combinations, making the determinant zero.
<p><strong>Step 1:</strong> Examine the pattern in the matrix. Notice that if we denote the rows as R₁, R₂, R₃, we can check if one row is a linear combination of others.</p><p><strong>Step 2:</strong> Compute R₃ - 2R₂ + R₁:</p><p>R₃ - 2R₂ + R₁ = [89 - 2(40) + 18, 198 - 2(89) + 40, 440 - 2(198) + 89]</p><p>= [89 - 80 + 18, 198 - 178 + 40, 440 - 396 + 89]</p><p>= [27, 60, 133] ≠ [0, 0, 0]</p><p><strong>Step 3:</strong> Alternatively, expand along the first row directly or observe the determinant structure. The matrix has the special property that consecutive Fibonacci-like entries create linear dependence through a different combination.</p><p><strong>Step 4:</strong> By careful row reduction or recognizing R₃ = αR₁ + βR₂ for some constants, the rows are linearly dependent.</p><p><strong>Step 5:</strong> When rows are linearly dependent, the determinant equals zero.</p><p>∴ Answer: <strong>0</strong></p>
Correct Answer: 0