1. Tardigrade
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  4. If A =(1/3)[1&2&2 2&1&-2 a&2&b] is an orthogonal matrix, then

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Q. If $A =\frac{1}{3}\begin{bmatrix}1&2&2\\ 2&1&-2\\ a&2&b\end{bmatrix}$ is an orthogonal matrix, then

Matrices

Solution:

As $A$ is an orthogonal matrix,
$A A^{T}=I$
$\Rightarrow \frac{1}{3}\begin{bmatrix}1&2&2\\ 2&1&-2\\ a&2&b\end{bmatrix}.\frac{1}{3} \begin{bmatrix}1&2&a\\ 2&1&2\\ 2&-2&b\end{bmatrix}=\begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&1\end{bmatrix}$
$\Rightarrow \frac{1}{9}\begin{bmatrix}1&2&2\\ 2&1&-2\\ a&2&b\end{bmatrix}.\frac{1}{3} \begin{bmatrix}1&2&a\\ 2&1&2\\ 2&-2&b\end{bmatrix}=\begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&1\end{bmatrix}$
$\Rightarrow \begin{bmatrix}9&0&a+4+2b\\ 0&9&2a+2-2b\\ a+4+2b&2a+2-2b&a^{2}+4+b^{2}\end{bmatrix}$
$=\begin{bmatrix}9&0&0\\ 0&9&0\\ 0&0&9\end{bmatrix}$
$\Rightarrow a+4+2 b=0,2 a+2-2 b=0 $ and $ a^{2}+4+b^{2}=9 $
$\Rightarrow a+2 b+4=0, a-b+1=0 $ and $ a^{2}+b^{2}=5$
$ \Rightarrow a=-2, b=-1 $