Matrices & Determinants
Determinants
Grade 12
Question:
<p>If matrix \(A\) is given by \(A = \begin{bmatrix} 6 & 11 \\ 2 & 4 \end{bmatrix}\), then the determinant of \(A^{2005} - 6A^{2004}\) is</p>
<p>\(2^{2006}\)</p>
<p>\((-11) \cdot 2^{2005}\)</p>
<p>\(-2^{2005} \cdot 7\)</p>
<p>\((-9) \cdot 2^{2004}\)</p>
Step-by-Step Solution
Key Concept: Factor out A^2004 from the expression A^2005 - 6A^2004 = A^2004(A - 6I), then use the property that det(AB) = det(A)·det(B) and det(kM) = k^n·det(M) for an n×n matrix.
<p><strong>Step 1:</strong> Factor the expression A²⁰⁰⁵ - 6A²⁰⁰⁴ = A²⁰⁰⁴(A - 6I)</p><p><strong>Step 2:</strong> Use the property det(AB) = det(A)·det(B):<br>det(A²⁰⁰⁴(A - 6I)) = det(A²⁰⁰⁴)·det(A - 6I)</p><p><strong>Step 3:</strong> Calculate det(A):<br>det(A) = (6)(4) - (11)(2) = 24 - 22 = 2<br>Therefore, det(A²⁰⁰⁴) = [det(A)]²⁰⁰⁴ = 2²⁰⁰⁴</p><p><strong>Step 4:</strong> Calculate det(A - 6I):<br>A - 6I = ⎡0 11⎤<br> ⎣2 -2⎦<br>det(A - 6I) = (0)(-2) - (11)(2) = 0 - 22 = -22</p><p><strong>Step 5:</strong> Combine the results:<br>det(A²⁰⁰⁴(A - 6I)) = 2²⁰⁰⁴ × (-22) = -22 × 2²⁰⁰⁴ = -11 × 2²⁰⁰⁵</p><p>∴ Answer: D</p>
Correct Answer: D