Matrices & Determinants
Matrix Powers
Grade 12
Question:
<p>If <span>w</span> ≠ 1 is the complex cube root of unity and matrix <span>H = <span style="font-family: Arial;">⎡</span><span style="font-family: Arial;">⎣</span>1 0<span style="font-family: Arial;">⎤</span><span style="font-family: Arial;">⎦</span></span><span style="font-family: Arial;">0 w</span>, then <span>H<sup>70</sup></span> is equal to</p>
<p>(A) H</p>
<p>(B) 0</p>
<p>(C) -H</p>
<p>(D) H²</p>
Step-by-Step Solution
Key Concept: Use the property that w³ = 1 for cube roots of unity to find the period of matrix powers
<p><strong>Step 1:</strong> Since w is a complex cube root of unity, we have w³ = 1.</p><p><strong>Step 2:</strong> Calculate H² = ⎡⎣1 0⎤⎦ · ⎡⎣1 0⎤⎦ = ⎡⎣1 0⎤⎦</p><p>0 w 0 w 0 w²</p><p><strong>Step 3:</strong> Calculate H³ = ⎡⎣1 0⎤⎦</p><p>0 w³ = ⎡⎣1 0⎤⎦ = I</p><p>0 1</p><p><strong>Step 4:</strong> Since H³ = I, we need to find H⁷⁰ = H^(3×23+1) = (H³)²³ · H = I²³ · H = H</p><p>∴ Answer is A.</p>
Correct Answer: A