Q. If a square matrix $A$ is such that $AA^{T} = I = A^{T}\ A\ then|A|$is equal to
A) 0
B) $\pm 1$
C) $\pm 2$
D) None of these
Explanation
(b)
Given, $A$ is a square matrix and $AA^{T} = I = A^{T}A$
$\Rightarrow \left| AA^{T} \right| = |I| = |A^{T}A|$
$\Rightarrow |A|\left| A^{T} \right| = I = \left| A^{T} \right||A|$
$\Rightarrow |A|^{2} =$1 [$\because\left| A^{T} \right| = |A|\rbrack$
$\Rightarrow |A| = \pm 1$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.