Question

If $A$ is an orthogonal matrix, then $A^{-1}$ equals

• A. $A^T$
• B. $A$
• C. $A^2$
• D. None of these

Answer 1

Answer: (A)

$A\cdot A^T=I$
$\Rightarrow \left|A\cdot A^T\right|=|I|$
$\Rightarrow |A{|}^2=1$
$\Rightarrow |A|=\pm 1$
$\Rightarrow A^{-1}$ exists
$\Rightarrow A^{-1}\cdot A\cdot A^T=A^{-1}\cdot I$
$\Rightarrow A^{-1}=A^T$