Question

Statement - 1 : if $|A|=\pm 1$, then the matrix $A$ is orthogonal matrix.
Statement - 2 : If $A$ is orthogonal matrix then if $|A|=\pm 1$

• A. Statement-1 is correct and Statement-2 is correct and Statement-2 is correct explanation of Statement-1
• B. Statement-1 is correct and Statement-2 is correct and Statement-2 is not the correct explanation of Statement-1
• C. Statement-1 is incorrect
• D. Statement-1=2 is incorrect

Answer 1

The correct option is C Statement-1 is incorrect

If $A$ is orthogonal matrix , then $A{A}^T=I$

If we apply det on both sides , we get $|A{|}^2=1$

$\Rightarrow |A|=\pm 1$

Therefore statement $2$ is correct

If $A$ is Involuntary matrix , then $A=A^{-1}$

If we apply det , we get $|A{|}^2=1$

$\Rightarrow |A|=\pm 1$

Therefore if $|A|+\pm$ , then matrix $A$ need not be orthogonal

Therefore statement $1$ is incorrect

So the correct option is $C$