If $A = \frac{1}{3}\begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b \end{bmatrix}$ is an orthogonal matrix of order 3, then:
Step-by-Step Solution
Key Concept: For an orthogonal matrix A, the condition AA^T = I means each row is a unit vector and rows are mutually orthogonal. With A = (1/3)M, this requires rows of M to have magnitude 3 and be perpendicular, giving orthogonality constraints: row·row = 9 and row_i·row_j = 0.
Given $AA^T = I$ for an orthogonal matrix, we set up the condition and multiply the given matrices. From $AA^T = I$, we get $9 = 0$ (contradiction check) and solve the system $a + 4 + 2b = 0$, $a - b + 1 = 0$, $a^2 + b^2 = 5$. This yields $a = -2, b = -1$.
Correct Answer: 1,3