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

Answer: (B)

The Given, A is a square matrix and $A{A}^T=I=A^{T}A$
$\Rightarrow$ $|A{A}^T|=|I|=|A^{T}A|$
$\Rightarrow$ $|A||A^T|=1=|A^T||A|$
$\Rightarrow$ $|A{|}^2=1$ $(\because |A^T|=|A|)$
$\Rightarrow$ $|A|=\pm 1.$