Question

If a square matrix A is such that $A{A}^T=I=A^{T}A$, then$|A|$is equal to

• A. $0$
• B. $\pm 1$
• C. $\pm 2$
• D. None of these

Answer 1

The correct option is B $\pm 1$

$Given$
$A{A}^T=I=A^{T}A$
$\Rightarrow \left|A{A}^T\right|=\left|I\right|=\left|A^{T}A\right|$
$\Rightarrow \left|A\right|\left|A^T\right|=1=\left|A^T\right|\left|A\right|$
$\Rightarrow {\left|A\right|}^2=1$
$\left|A^T\right|=\left|A\right|$
$\Rightarrow \left|A\right|=\pm 1$