Question

If $n^{\text{th}}-$ order square matrix $A$ is orthogonal, then, $|adj\left(adjA\right)|$ is

• A. always $-1$ if $n$ is even
• B. always $1$ if $n$ is odd
• C. always $1$
• D. None of the above

Answer 1

The correct option is B always $1$ if $n$ is odd

Since $A$ is orthogonal
$\Rightarrow A{A}^T=I\Rightarrow \left|A^2\right|=1\Rightarrow |A|=\pm 1$
Now, $|adj\left(adjA\right)|=|A{|}^{(n-1{)}^2}$
$={|\pm 1|}^{(n-1{)}^2}$
$=1$ only when $n$ is odd