Question

Evaluate det (A) by expanding along the 2nd row.

Answer 1

$$det(A)=\sum_{j=1}^n (-1)^{i+j} a_{ij} M_j^i$$

$M_j^i - additional \space minor \space to \space element \space a_{ij}$

$$det(A)=\begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix} =(-1)^{2+1}a_{21}\begin{vmatrix} a_{12} & a_{13} \\ a_{32} & a_{33} \end{vmatrix}+$$

$$(-1)^{2+2}a_{22}\begin{vmatrix} a_{11} & a_{13} \\ a_{31} & a_{33} \end{vmatrix}+(-1)^{2+3}a_{23}\begin{vmatrix} a_{11} & a_{12} \\ a_{31} & a_{32} \end{vmatrix}=$$

$$-a_{21}(a_{12}a_{33}-a_{13}a_{32})+a_{22}(a_{11}a_{33}-a_{13}a_{31})-a_{23}(a_{11}a_{32}-a_{12}a_{31})=$$

$$-a_{12}a_{21}a_{33}+a_{13}a_{21}a_{32}+a_{11}a_{22}a_{33}-a_{13}a_{22}a_{31}-a_{11}a_{23}a_{32}+a_{12}a_{23}a_{31}$$