If $3 A = [\begin{array}{ccc} 1 & 2 & 2 \\ 2 & 1 & - 2 \\ a & 2 & b \end{array}]$ and A is an orthogonal matrix then the product of a and b is equal to

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answer is 4.

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Detailed Solution

$\begin{array}{l} {AA}^{T} = I \Rightarrow \frac{1}{9} |\begin{array}{ccc} 1 & 2 & 2 \\ 2 & 1 & - 2 \\ a & 2 & b \end{array}| |\begin{array}{ccc} 1 & 2 & a \\ 2 & 1 & 2 \\ 2 & - 2 & b \end{array}| = |\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}| or \\ \frac{1}{9} |\begin{array}{ccc} 9 & 0 & a + 4 + 2 b \\ 0 & 9 & 2 a + 2 - 2 b \\ a + 4 + 2 b & 2 a + 2 - 2 b & a^{2} + 4 + b^{2} \end{array}| = |\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}| \end{array}$