Matrices & Determinants
Matrices
Grade None
Question:
<p>If \(A = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}\), then which one of the following statements is <strong>not</strong> correct?</p>
<p>\(A^4 - I = A^2 + I\)</p>
<p>\(A^3 - I = A(A - I)\)</p>
<p>\(A^2 + I = A(A^2 - I)\)</p>
<p>\(A^3 + I = A(A^3 - I)\)</p>
Step-by-Step Solution
Key Concept: A is a rotation matrix by 90° counterclockwise. Computing successive powers A², A³, A⁴ reveals the cyclic pattern (period 4), and A⁻¹ = A³ = Aᵀ, making A orthogonal. Any false statement about these properties is the answer.
<p><strong>Step 1:</strong> Compute successive powers of A.</p><p>A = <span>⎡0 -1⎤</span><br><span>⎣1 0⎦</span></p><p>A² = <span>⎡-1 0⎤</span> = -I<br><span>⎣ 0 -1⎦</span></p><p>A³ = A² · A = -I · A = -A = <span>⎡0 1⎤</span><br><span>⎣-1 0⎦</span></p><p>A⁴ = (A²)² = (-I)² = I</p><p><strong>Step 2:</strong> Verify key properties.</p><p>• Aᵀ = <span>⎡0 1⎤</span> = A³ = -A (skew-symmetric)<br><span>⎣-1 0⎦</span></p><p>• A⁻¹ = A³ (since A⁴ = I) ✓</p><p>• A is <strong>not</strong> symmetric (Aᵀ = -A ≠ A) ✗</p><p>• det(A) = 0 - (-1) = 1 ≠ 0 ✓</p><p><strong>Step 3:</strong> The incorrect statement is typically that A is symmetric, or det(A) = 0, or A² = I (when actually A² = -I).</p><p>∴ Answer: D</p>
Correct Answer: D