<p>If \(A = \begin{bmatrix} -4 & -4 \\ 3 & 1 \end{bmatrix}\), then the determinant of the matrix \((A^{2016} - 2A^{2015} - A^{2014})\) is</p>
Step-by-Step Solution
Key Concept: Find the characteristic equation of A to establish a recurrence relation (Cayley-Hamilton theorem), then factor out powers of A to reduce the complex expression to a manageable form involving det(A).
<p><strong>Step 1:</strong> Find det(A) and trace(A) to use Cayley-Hamilton theorem.</p><p>det(A) = (-4)(1) - (-4)(3) = -4 + 12 = 8</p><p>tr(A) = -4 + 1 = -3</p><p><strong>Step 2:</strong> By Cayley-Hamilton theorem: A² + 3A - 8I = 0, so A² = -3A + 8I</p><p><strong>Step 3:</strong> Factor the given expression:</p><p>A²⁰¹⁶ - 2A²⁰¹⁵ - A²⁰¹⁴ = A²⁰¹⁴(A² - 2A - I)</p><p><strong>Step 4:</strong> Substitute the characteristic equation A² = -3A + 8I:</p><p>A² - 2A - I = (-3A + 8I) - 2A - I = -5A + 7I</p><p><strong>Step 5:</strong> Therefore:</p><p>det(A²⁰¹⁶ - 2A²⁰¹⁵ - A²⁰¹⁴) = det(A²⁰¹⁴) · det(-5A + 7I)</p><p>det(A²⁰¹⁴) = [det(A)]²⁰¹⁴ = 8²⁰¹⁴</p><p><strong>Step 6:</strong> For det(-5A + 7I):</p><p>-5A + 7I = <span style='border: 1px solid;'> 13 -20 <br/> -15 2 </span></p><p>det(-5A + 7I) = (13)(2) - (-20)(-15) = 26 - 300 = -274</p><p><strong>Step 7:</strong> Final answer: det = 8²⁰¹⁴ · (-274) = -274 · 8²⁰¹⁴</p><p>∴ Answer: <strong>D</strong></p>
Correct Answer: D