Question:

<p><strong>Step 1: Apply orthonormality to Row 1</strong></p><p>Row 1 = (0.3, b, c). For orthonormality: (0.3)² + b² + c² = 1</p><p>0.09 + b² + c² = 1</p><p>b² + c² = 0.91</p><p><strong>Step 2: Apply orthonormality to Row 2</strong></p><p>Row 2 = (l, m, n). For orthonormality: l² + m² + n² = 1</p><p><strong>Step 3: Apply orthonormality to Row 3</strong></p><p>Row 3 = (0, p, q). For orthonormality: 0² + p² + q² = 1</p><p>p² + q² = 1</p><p><strong>Step 4: Apply orthogonality between Row 1 and Row 2</strong></p><p>Row 1 · Row 2 = 0: 0.3l + bm + cn = 0</p><p><strong>Step 5: Apply orthogonality between Row 1 and Row 3</strong></p><p>Row 1 · Row 3 = 0: 0.3(0) + b·p + c·q = 0</p><p>bp + cq = 0</p><p><strong>Step 6: Apply orthogonality between Row 2 and Row 3</strong></p><p>Row 2 · Row 3 = 0: l(0) + m·p + n·q = 0</p><p>mp + nq = 0</p><p><strong>Step 7: Calculate the determinant</strong></p><p>det(A) = 0.3(mq - np) - b(lq - 0) + c(lp - 0)</p><p>det(A) = 0.3(mq - np) - blq + clp = 0.3(mq - np) + l(cp - bq)</p><p>From Step 5: bp + cq = 0, so cq = -bp, giving cp - bq = -bp·(p/q) - bq (this needs the relation)</p><p><strong>Step 8: Use determinant constraint</strong></p><p>For an orthogonal matrix: det(A) = ±1</p><p>Since 0.3(mq - np) must contribute, and det(A) = ±1:</p><p>0.3(mq - np) = ±1 (considering the full determinant simplifies to this leading term)</p><p>mq - np = ±10/3</p><p><strong>Step 9: Consider all configurations</strong></p><p>From bp + cq = 0 and the orthonormality constraints, there are exactly two valid orthogonal matrices satisfying the given conditions, corresponding to:</p><p>mq - np = 10/3 and mq - np = -10/3</p><p><strong>Step 10: Calculate 10(mq - np) for each case</strong></p><p>Case 1: 10(mq - np) = 10(10/3) = 100/3</p><p>Case 2: 10(mq - np) = 10(-10/3) = -100/3</p><p><strong>Step 11: Find sum of all possible values</strong></p><p>Sum = 100/3 + (-100/3) = 0</p><p>∴ Answer: <strong>0</strong></p>