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  4. If A is an orthogonal matrix, then A-1 equals

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

Matrices

Solution:

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