<p>If <em>A</em> = \(\begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b \end{bmatrix}\) is a matrix satisfying the equation \(AA^T = 9I\), where <em>I</em> is \(3 \times 3\) identity matrix, then the ordered pair \((a, b)\) is equal to</p>
Step-by-Step Solution
Key Concept: For AA^T = 9I, the rows of A must be orthogonal (dot product = 0) and each row must have magnitude √9 = 3. This creates a system of equations from row orthogonality and row norm conditions.
<p><strong>Step 1:</strong> Write the condition AA^T = 9I explicitly. The rows of A must satisfy: Row_i · Row_j = 9δ_ij (where δ_ij is Kronecker delta).</p><p><strong>Step 2:</strong> For Row 1: 1² + 2² + 2² = 1 + 4 + 4 = 9 ✓</p><p><strong>Step 3:</strong> For Row 2: 2² + 1² + (-2)² = 4 + 1 + 4 = 9 ✓</p><p><strong>Step 4:</strong> For Row 3: a² + 4 + b² = 9, so a² + b² = 5 ... (i)</p><p><strong>Step 5:</strong> Row 1 · Row 2 = 1(2) + 2(1) + 2(-2) = 2 + 2 - 4 = 0 ✓</p><p><strong>Step 6:</strong> Row 1 · Row 3 = 1(a) + 2(2) + 2(b) = a + 4 + 2b = 0, so a + 2b = -4 ... (ii)</p><p><strong>Step 7:</strong> Row 2 · Row 3 = 2(a) + 1(2) + (-2)(b) = 2a + 2 - 2b = 0, so a - b = -1 ... (iii)</p><p><strong>Step 8:</strong> From (iii): a = b - 1. Substituting in (ii): (b - 1) + 2b = -4 → 3b = -3 → b = -1, so a = -2</p><p><strong>Step 9:</strong> Verify with (i): (-2)² + (-1)² = 4 + 1 = 5 ✓</p><p>∴ Answer: (a, b) = (-2, -1) or D</p>
Correct Answer: D