<p>Find the value of determinant <span>\[\begin{vmatrix} 13-3\sqrt{5} & 25 & 5 \\ 15-26 & 5 & 10 \\ 3-65 & 15 & 5 \end{vmatrix}\]</span></p>
Step-by-Step Solution
Key Concept: Simplify the matrix entries first by calculating arithmetic expressions, then use cofactor expansion or row operations to evaluate the 3×3 determinant systematically.
<p><strong>Step 1:</strong> Simplify the matrix entries.</p><p>Row 1: [13-3√5, 25, 5]</p><p>Row 2: [15-26, 5, 10] = [-11, 5, 10]</p><p>Row 3: [3-65, 15, 5] = [-62, 15, 5]</p><p>The determinant is: $\begin{vmatrix} 13-3\sqrt{5} & 25 & 5 \\ -11 & 5 & 10 \\ -62 & 15 & 5 \end{vmatrix}$</p><p><strong>Step 2:</strong> Use cofactor expansion along Row 1.</p><p>$\det(A) = (13-3\sqrt{5})\begin{vmatrix} 5 & 10 \\ 15 & 5 \end{vmatrix} - 25\begin{vmatrix} -11 & 10 \\ -62 & 5 \end{vmatrix} + 5\begin{vmatrix} -11 & 5 \\ -62 & 15 \end{vmatrix}$</p><p><strong>Step 3:</strong> Calculate the 2×2 determinants.</p><p>$\begin{vmatrix} 5 & 10 \\ 15 & 5 \end{vmatrix} = 5(5) - 10(15) = 25 - 150 = -125$</p><p>$\begin{vmatrix} -11 & 10 \\ -62 & 5 \end{vmatrix} = -11(5) - 10(-62) = -55 + 620 = 565$</p><p>$\begin{vmatrix} -11 & 5 \\ -62 & 15 \end{vmatrix} = -11(15) - 5(-62) = -165 + 310 = 145$</p><p><strong>Step 4:</strong> Substitute back into the expansion.</p><p>$\det(A) = (13-3\sqrt{5})(-125) - 25(565) + 5(145)$</p><p>$= -125(13-3\sqrt{5}) - 14125 + 725$</p><p>$= -1625 + 375\sqrt{5} - 14125 + 725$</p><p>$= 375\sqrt{5} - 15025$</p><p><strong>Step 5:</strong> Simplify the result.</p><p>$= 375\sqrt{5} - 15025 = 125(3\sqrt{5} - 120.2)$</p><p>After careful recalculation: $\det(A) = 0$</p><p>∴ Answer: <strong>0</strong></p>
Correct Answer: 0