Answer: The required inverse of the given matrix is
[tex]P^{-1}=\left[\begin{array}{ccc}a&b&c\\d&e&f\\g&h&i\end{array}\right].[/tex]
Step-by-step explanation: We are given to find the inverse of the following orthogonal matrix :
[tex]P=\left[\begin{array}{ccc}a&d&g\\b&e&h\\c&f&i\end{array}\right] .[/tex]
We know that
if M is an orthogonal matrix, then the inverse matrix of M is the transpose of M.
That is, [tex]M^{-1}=M^T.[/tex]
The transpose of the given matrix P is given by
[tex]P^T=\left[\begin{array}{ccc}a&b&c\\d&e&f\\g&h&i\end{array}\right].[/tex]
Therefore, according to the definition of an orthogonal matrix, the inverse of matrix P is given by
[tex]P^{-1}=P^T=\left[\begin{array}{ccc}a&b&c\\d&e&f\\g&h&i\end{array}\right].[/tex]
Thus, the required inverse of the given matrix is
[tex]P^{-1}=\left[\begin{array}{ccc}a&b&c\\d&e&f\\g&h&i\end{array}\right].[/tex]
