<p>If \(A = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b \end{bmatrix}\) is a matrix satisfying the equation \(AA^T = 9I\), where \(I\) 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, each row of A must have magnitude √9 = 3, and distinct rows must be orthogonal. Use row orthogonality and the magnitude constraint simultaneously to find a and b.
<p><strong>Step 1: Set up the AA^T equation</strong></p><p>AA^T = 9I means: (1) each row has squared magnitude 9, and (2) different rows are orthogonal.</p><p><strong>Step 2: Apply magnitude constraint to row 3</strong></p><p>a² + 4 + b² = 9, so a² + b² = 5</p><p><strong>Step 3: Apply orthogonality between rows 1 and 3</strong></p><p>Row 1 · Row 3 = 0: 1(a) + 2(2) + 2(b) = 0</p><p>a + 2b + 4 = 0, so a + 2b = -4</p><p><strong>Step 4: Apply orthogonality between rows 2 and 3</strong></p><p>Row 2 · Row 3 = 0: 2(a) + 1(2) + (-2)(b) = 0</p><p>2a - 2b + 2 = 0, so 2a - 2b = -2, giving a - b = -1</p><p><strong>Step 5: Solve the system</strong></p><p>From a - b = -1: a = b - 1</p><p>Substitute into a + 2b = -4: (b-1) + 2b = -4, so 3b = -3, thus b = -1</p><p>Then a = -1 - 1 = -2</p><p><strong>Step 6: Verify</strong></p><p>Check a² + b² = 4 + 1 = 5 ✓</p><p>∴ Answer: (a,b) = (-2, -1)</p>
Correct Answer: D