<p>Let \(A = \begin{bmatrix}i & 0\\0 & i\end{bmatrix}\). Find the trace of \(A^{4n}\).</p>
Step-by-Step Solution
Key Concept: Recognize that A is a scalar multiple of the identity matrix (A = iI), so A^(4n) = (iI)^(4n) = i^(4n)·I. Since i^4 = 1, we have i^(4n) = (i^4)^n = 1^n = 1, making A^(4n) = I.
<p><strong>Step 1:</strong> Recognize the structure of A</p><p>A = iI, where I is the 2×2 identity matrix. This is a scalar multiple of the identity.</p><p><strong>Step 2:</strong> Compute A^(4n)</p><p>A^(4n) = (iI)^(4n) = i^(4n) · I^(4n) = i^(4n) · I</p><p><strong>Step 3:</strong> Simplify i^(4n)</p><p>Since i⁴ = (i²)² = (-1)² = 1, we have:</p><p>i^(4n) = (i⁴)ⁿ = 1ⁿ = 1</p><p><strong>Step 4:</strong> Find the trace</p><p>A^(4n) = 1 · I = I = ⎡1 0⎤</p><p> ⎣0 1⎦</p><p>trace(A^(4n)) = trace(I) = 1 + 1 = <strong>2</strong></p><p>∴ Answer: B</p>
Correct Answer: B