Let A be an orthogonal square matrix.
Statement-1: $A^{-1}$ is an orthogonal matrix.
Statement-2: $(A^{-1})^T=(A^T)^{-1}$ and $(AB)^{-1}=B^{-1} A^{-1}$
Answer & explanation
Correct answer: option 1
Clearly, statement-2 is true. Since A is an orthogonal matrix.
$∴AA^T = A^TA =I$
$⇒(AA^T)^{-1}=(A^TA)^{-1} = I$
$⇒(A^T) A^{-1}=A^{-1} (A^T)^{-1}$ $[∵ (AB)^{-1} = B^{-1} A^{-1}]$
$⇒(A^{-1})^T A^{-1}=A^{-1} (A^{-1})^T$ $[∵ (A^T)^{-1}=(A^{-1})^T]$
$⇒A^{-1}$ is orthogonal.