Question

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

• A. always $-1$ if $n$ is even
• B. always $1$ if $n$ is odd
• C. always $1$
• D. none of these

Answer 1

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

Given $A$ is a $n^{th}$ order square, orthogonal matrix.
$\Rightarrow A{A}^T=I$
$\Rightarrow |A{|}^2=1$
$\Rightarrow |A|=\pm 1$
Now, $|adj\left(adjA\right)|=|A{|}^{(n-1{)}^2}$
$(n-1{)}^2$ is even, when n is odd.
$\therefore |adj\left(adjA\right)|=|A{|}^{(n-1{)}^2}$ is always $1$ if $n$ is odd.
Hence, option B.