Matrices & Determinants
Orthogonal Matrix
Grade 12
Question:
<p>If \(A = \begin{bmatrix}1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b\end{bmatrix}\) and \(A \cdot A^T = 9I\), then find the values of \(a\) and \(b\).</p>
<p>\(a = -2,\ b = -1\)</p>
<p>\(a = 1,\ b = -2\)</p>
<p>\(a = -2,\ b = 1\)</p>
<p>\(a = 2,\ b = -1\)</p>
Step-by-Step Solution
Key Concept: When AA^T = 9I, the rows of A must be orthogonal vectors with magnitude 3 each. Use the orthogonality condition (row_i · row_j = 0 for i≠j) and magnitude condition (row_i · row_i = 9) to find unknowns.
<p><strong>Step 1:</strong> Set up the orthogonality conditions. For AA^T = 9I, rows must satisfy:</p><p>• Row 1 · Row 1 = 9: 1² + 2² + 2² = 1 + 4 + 4 = 9 ✓</p><p>• Row 2 · Row 2 = 9: 2² + 1² + (-2)² = 4 + 1 + 4 = 9 ✓</p><p>• Row 3 · Row 3 = 9: a² + 4 + b² = 9, so <strong>a² + b² = 5</strong> ... (1)</p><p><strong>Step 2:</strong> Apply orthogonality between different rows (dot product = 0):</p><p>• Row 1 · Row 2 = 0: (1)(2) + (2)(1) + (2)(-2) = 2 + 2 - 4 = 0 ✓</p><p>• Row 1 · Row 3 = 0: (1)(a) + (2)(2) + (2)(b) = a + 4 + 2b = 0, so <strong>a + 2b = -4</strong> ... (2)</p><p>• Row 2 · Row 3 = 0: (2)(a) + (1)(2) + (-2)(b) = 2a + 2 - 2b = 0, so <strong>a - b = -1</strong> ... (3)</p><p><strong>Step 3:</strong> Solve the system. From (3): a = b - 1. Substitute into (2):</p><p>(b - 1) + 2b = -4 → 3b = -3 → <strong>b = -1</strong></p><p>Therefore: <strong>a = -2</strong></p><p><strong>Step 4:</strong> Verify with equation (1): (-2)² + (-1)² = 4 + 1 = 5 ✓</p><p>∴ Answer: <strong>a = -2, b = -1</strong></p>
Correct Answer: C