<p>Find the value of \(x\) for which the matrix \[A = \begin{bmatrix} 2/x & -1 & 2 \\ 1 & x & 2x^2 \\ 1 & 1/x & 2 \end{bmatrix}\] is singular.</p>
Step-by-Step Solution
Key Concept: A matrix is singular when its determinant equals zero. Express det(A) as a function of x and solve det(A) = 0 by expanding along a convenient row or column.
<p><strong>Step 1:</strong> For matrix A to be singular, det(A) = 0.</p><p><strong>Step 2:</strong> Expand det(A) along the first row:</p><p>det(A) = (2/x)[x·2 - 2x²·(1/x)] - (-1)[1·2 - 2x²·1] + 2[1·(1/x) - x·1]</p><p><strong>Step 3:</strong> Simplify each minor:</p><p>• First term: (2/x)[2x - 2x] = (2/x)[0] = 0</p><p>• Second term: +1[2 - 2x²] = 2 - 2x²</p><p>• Third term: 2[1/x - x] = 2/x - 2x</p><p><strong>Step 4:</strong> Combine:</p><p>det(A) = 0 + (2 - 2x²) + (2/x - 2x) = 0</p><p><strong>Step 5:</strong> Multiply through by x (x ≠ 0):</p><p>2x - 2x³ + 2 - 2x² = 0</p><p>-2x³ - 2x² + 2x + 2 = 0</p><p>x³ + x² - x - 1 = 0</p><p><strong>Step 6:</strong> Factor by grouping:</p><p>x²(x + 1) - 1(x + 1) = 0</p><p>(x + 1)(x² - 1) = 0</p><p>(x + 1)(x - 1)(x + 1) = 0</p><p>(x + 1)²(x - 1) = 0</p><p><strong>Step 7:</strong> Solutions are x = -1 (double root) and x = 1.</p><p>∴ Answer: <strong>x = ±1</strong></p>
Correct Answer: ±1