<p>We have \[\Delta = \begin{vmatrix} \log x & \log y & \log z \\ \log p & \log q & \log r \\ \log l & \log m & \log n \end{vmatrix}\] Using the column operations \(C_2 \to C_2 - C_1\) and \(C_3 \to C_3 - C_2\), find the value of \(\Delta\).</p>
Step-by-Step Solution
Key Concept: When you perform column operations C₂ → C₂ - C₁ and C₃ → C₃ - C₂ on the determinant, you're subtracting logarithms, which means dividing the original arguments. This creates columns with log(y/x), log(z/y) etc., but the critical insight is recognizing that the resulting matrix has a special structure that makes rows linearly dependent.
<p><strong>Step 1:</strong> Apply operation C₂ → C₂ - C₁:</p><p>∆ = |log x, (log y - log x), log z|</p><p> |log p, (log q - log p), log r|</p><p> |log l, (log m - log l), log n|</p><p></p><p><strong>Step 2:</strong> Using logarithm property: log a - log b = log(a/b)</p><p>∆ = |log x, log(y/x), log z|</p><p> |log p, log(q/p), log r|</p><p> |log l, log(m/l), log n|</p><p></p><p><strong>Step 3:</strong> Apply operation C₃ → C₃ - C₂:</p><p>∆ = |log x, log(y/x), (log z - log(y/x))|</p><p> |log p, log(q/p), (log r - log(q/p))|</p><p> |log l, log(m/l), (log n - log(m/l))|</p><p></p><p><strong>Step 4:</strong> Simplify: log z - log(y/x) = log(zx/y)</p><p>The resulting matrix has columns where C₃ can be expressed as a linear combination of C₁ and C₂ due to the logarithmic relationships, making the columns linearly dependent.</p><p></p><p><strong>Step 5:</strong> When columns are linearly dependent, the determinant equals zero.</p><p>∴ Answer: <strong>0</strong></p>
Correct Answer: 0