If $P$ is an orthogonal matrix and $Q=PA{{P}^{T}}$ and $B={{P}^{T}}{{Q}^{1000}}P$, then ${{B}^{-1}}$ is, where $A$ is involuntary matrix:
(A) $A$
(B) ${{A}^{1000}}$
(C) $I$
(D) None of these

Hint: For answering this question we will simplify the value of the matrix $B$ using the given information as $P$ is an orthogonal matrix, $P{{P}^{T}}=I$ and $A$ is an involuntary matrix so ${{A}^{2}}=I$ . Using this we will simplify and find the value of ${{B}^{-1}}$ .

Complete step by step answer:
From the definition for an orthogonal matrix, the product of a matrix and its transpose is equal to an identity matrix. The definition of an involuntary matrix states that the square of the matrix is equal to the identity matrix.
Now considering from the basic definition, we have $P$ is an orthogonal matrix that implies that $P{{P}^{T}}=I\Rightarrow {{P}^{-1}}={{P}^{T}}$ and $A$ is involuntary matrix that implies that ${{A}^{2}}=I\Rightarrow A={{A}^{-1}}$ .
We have from the question that $Q=PA{{P}^{T}}$ as $P$ is an orthogonal matrix we can simply write it as $Q=A$ .
We have from the question that $B={{P}^{T}}{{Q}^{1000}}P$ as $P$ is orthogonal matrix we can simply writes it as $B={{Q}^{1000}}$ .
As we have $Q=A$ we can say $B={{A}^{1000}}$ .
So we can say that from ${{A}^{2}}=I$ , $B={{\left( {{A}^{2}} \right)}^{500}}={{I}^{500}}=I$
In the question it has been asked for the value of ${{B}^{-1}}$ we can say that from $B=I$ that ${{B}^{-1}}=I$ .
Hence we can conclude that when $P$ is an orthogonal matrix and $Q=PA{{P}^{T}}$ and $B={{P}^{T}}{{Q}^{1000}}P$, then ${{B}^{-1}}=I$ , where $A$ is involuntary matrix.

So, the correct answer is “Option C”.

Note: While answering this type of questions we should take care that when $B={{A}^{1000}}$ it can be further simplified if we forget that then we will have ${{B}^{-1}}$ as ${{A}^{1000}}$ which is a complete wrong answer. This will lead us to a wrong conclusion.