Each element of a real orthogonal matrix is equal to its cofactor

Question

If $A =(a_{ij})$ be a real orthogonal matrix with $\det A = 1$, prove that each element $a_{rs}$ of $A$ is equal to its cofactor $A_{rs}$ in $\det A$. I got this basic problem from my text book and somehow I couldn't proceed, please help me to get this result.

Answer 1

$A^t = A^{-1} = (\operatorname{cofactor}(A))^t $ implies $A = \operatorname{cofactor}(A)$ and I am not sure why $A$ should have determinant $1$. It seems the conclusion is true for any real orthogonal matrix.

Answer 2

$A^t = A^{-1} = \frac{1}{\det A}(\operatorname{cofactor}(A))^t\quad$ implies $\quad A = \operatorname{cofactor}(A)\quad$ only if $\quad\det A=1$

The condition $\det A = 1$ is needed, since the determinant appears on the inverse matrix formula. For a general real orthogonal matrix you would get

$A = \frac{1}{\det A}\operatorname{cofactor}(A)$

so each element would be proportional to its cofactor, with $\frac{1}{\det A}$ being the ratio.