Determinants

Expanding along \(R_{2}\) we get \(|A|=|\begin{smallmatrix}a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33}\end{smallmatrix}|\) \(|A|=(-1)^{2+1}a_{21}|\begin{smallmatrix}a_{12}&a_{13}\\ a_{32}&a_{33}\end{smallmatrix}|+(-1)^{2+2}a_{22}|\begin{smallmatrix}a_{11}&a_{13}\\ a_{31}&a_{33}\end{smallmatrix}| +(-1)^{2+3}a_{23}|\begin{smallmatrix}a_{11}&a_{12}\\ a_{31}&a_{32}\end{smallmatrix}|\) \(=-a_{21}(a_{12}a_{33}-a_{32}a_{13})+a_{22}(a_{11}a_{33}-a_{31}a_{13})\) \(-a_{23}(a_{11}a_{32}-a_{31}a_{12})\) \(|A|=-a_{21}a_{12}a_{33}+a_{21}a_{32}a_{13}+a_{22}a_{11}a_{33}-a_{22}a_{31}a_{13}-a_{23}a_{11}a_{32}\) + A23 A31 A12 \(=a_{11}a_{22}a_{33}-a_{11}a_{23}a_{32}-a_{12}a_{21}a_{33}+a_{12}a_{23}a_{31}+a_{13}a_{21}a_{32}\) \(-a_{13}a_{31}a_{22}\)