Determinants

By expanding along \(C_{1}\) we get \(|A|=|\begin{smallmatrix}a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33}\end{smallmatrix}|\) \(|A|=a_{11}(-1)^{1+1}|\begin{smallmatrix}a_{22}&a_{23}\\ a_{32}&a_{33}\end{smallmatrix}|+a_{21}(-1)^{2+1}|\begin{smallmatrix}a_{12}&a_{13}\\ a_{32}&a_{33}\end{smallmatrix}| +a_{31}(-1)^{3+1}|\begin{smallmatrix}a_{12}&a_{13}\\ a_{22}&a_{23}\end{smallmatrix}|\) \(=a_{11}(a_{22}a_{33}-a_{23}a_{32})-a_{21}(a_{12}a_{33}-a_{13}a_{32})+a_{31}(a_{12}a_{23}-a_{13}a_{22})\) \(|A|=a_{11}a_{22}a_{33}-a_{11}a_{23}a_{32}-a_{21}a_{12}a_{33}+a_{21}a_{13}a_{32}+a_{31}a_{12}a_{23}\) \(-a_{13}a_{31}a_{22}\) \(-a_{31}a_{13}a_{22}\) \(=a_{11}a_{22}a_{33}-a_{11}a_{23}a_{32}-a_{12}a_{21}a_{33}+a_{12}a_{23}a_{31}+a_{13}a_{21}a_{32}\) (3)

Clearly, values of A in (1), (2) and (3) are equal. It is left as an exercise to the reader to verify that the values of |A| by expanding along \(R_{3}\), \(C_{2}\) and \(C_{3}\) are equal to the value of \(|A|\) obtained in (1), (2) or (3).

Hence, expanding a determinant along any row or column gives same value.