Q. If $A$ is a non-singular matrix such that $AA^{T} = A^{T}A$ and $B = A^{- 1}A^{T}$, then matrix $B$ is
A) Involuntary
B) Orthogonal
C) Idempotent
D) None of these
Explanation
(b)
Given,
$B = A^{- 1}A^{T}$
$\Rightarrow B^{T} = \left( A^{- 1}A^{T} \right) = A \times \left( A^{- 1} \right)^{T}$
$\Rightarrow B \times B^{T} = A^{- 1}A^{T} \times A \times \left( A^{- 1} \right)^{T} = A^{- 1} \times \left( A^{T} \times A \right)\left( A^{- 1} \right)^{T}$
$= A^{- 1}\left( A \times A^{T} \right)\left( A^{- 1} \right)^{T}$
$= \left( A^{- 1}A^{} \right) \times \left( A^{- 1}A \right)^{T} = I \times I^{T} = I$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.