Question

If A is an orthogonal matrix and $B=AP$ where P is a non singular matrix then the matrix $P{B}^{-1}$ is also orthogonal.
If true enter 1 else enter 0

Answer 1

Given $A$ is an orthogonal matrix and $B=AP$ where $P$ is a non singular matrix.

$\Rightarrow A{A}^T=A^{T}A=I$

$P{B}^{-1}{\left(P{B}^{-1}\right)}^T=P{\left(AP\right)}^{-1}\left({\left(B^{-1}\right)}^T{P}^T\right)$

$=P{P}^{-1}{A}^{-1}{\left(P^{-1}{A}^{-1}\right)}^T{P}^T$

$=A^{-1}{\left(A^{-1}\right)}^T{\left(P^{-1}\right)}^T{P}^T$

$==A^{-1}{\left(A^T\right)}^{-1}{\left(P^T\right)}^{-1}{P}^T$

$={\left(A^{T}A\right)}^{-1}{\left(P^T\right)}^{-1}{P}^T$

$=I^{-1}I=I$

$\therefore P{B}^{-1}$ is an orthogonal matrix.

Hence, the answer is $1$.