Question

If $A$ is an orthogonal matrix of order $n$, then $|\text{adj}\left(\text{adj}\right(A\left)\right)|$ is

• A. $1,$ when $n=4$
• B. $-1,$ when $n=6$
• C. $-1,$ when $n=7$
• D. $1,$ when $n=5$

Answer 1

Answer: (A), (B), (D)

$1,$ when $n=5$

Since $A$ is orthogonal
$\Rightarrow A{A}^T=I\Rightarrow {\left|A\right|}^2=1\Rightarrow |A|=\pm 1$
Now, $|\text{adj}\left(\text{adj}\right(A\left)\right)|=|A{|}^{(n-1{)}^2}[\because \text{adj}(\text{adj}\left(A\right))=|A{|}^{n-2}A]=|A{|}^{(n-1{)}^2}$
$=\{\begin{array}{cc}\pm 1,& n\in \text{even}\\ 1,& n\in \text{odd}\end{array}$