Question

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

Explanation for the correct option

Given that $n^{th}$ order square matrix $A$ is a orthogonal

$$\begin{gathered} \Rightarrow A^T=A^{-1} \\ \Rightarrow A.A^T=I \end{gathered}$$

We know that the value of the determinant of an orthogonal matrix is $\pm 1$

$\Rightarrow \left|A\right|=\pm 1$

We know that $\left|adj\left(adj\left(A\right)\right)\right|={\left|A\right|}^{{\left(n-1\right)}^2}$

If $n$is odd then ${\left(n-1\right)}^2$ is even.

$\Rightarrow {\left|A\right|}^{even}={\left(\pm 1\right)}^{even}=1$always

Hence the correct option is option(B) i.e. always $1$ if $n$is odd