Question

If A is an orthogonal matrix and $|A|=-1,$ then $A^T$ is equal to

• A. $-A$
• B. $A$
• C. $-\text{adj}\left(A\right)$
• D. $\text{adj}\left(A\right)$

Answer 1

The correct option is C $-\text{adj}\left(A\right)$

Here given that $|A|=-1$

For an orthogonal matrix, $A{A}^T=I$
$\Rightarrow A^T=A^{-1}$
$=\frac{\text{adj}\left(A\right)}{|A|}=-\text{adj}\left(A\right)$