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, $AA^T = I$ provides constraints that determine all unknown entries.
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