Question

Statement - 1 : If $A$ is an orthogonal matrix of order $n$, then $\left|adj\left(adjA\right)\right|=\pm 1$
Statement - 2 : $\left|adj\left(adjA\right)\right|={\left|A\right|}^{n^2-1}$

• A. Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1
• B. Statement-1 is True, Statement-2 is True, Statement-2 is NOT a correct explanation for Statement-1
• C. Statement-1 is True, Statement-2 is False.
• D. Statement-1 is False, Statement-2 is True.

Answer 1

The correct option is C Statement-1 is True, Statement-2 is False.

Since,A is an orthogonal matrix
$A{A}^{\prime }=A^{\prime }A=I$
$\Rightarrow |A{A}^{\prime }|=1$
$\Rightarrow |A||A^{\prime }|=1$
$\Rightarrow |A{|}^2=1(\because |A|=|A^T|)$
$\Rightarrow |A|=\pm 1$
Now, $|adj\left(adjA\right)=|A{|}^{(n-1{)}^2}$
$=(\pm 1{)}^{(n-1{)}^2}$
$=\pm 1$
Hence, statement 1 is true.
Statement 2 is false.