Find minors and cofactors of the elements $a_{11}$, $a_{21}$ in the determinant $\Delta = \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix}$.
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → $M_{11} = a_{22}a_{33} - a_{23}a_{32}, \quad A_{11} = a_{22}a_{33} - a_{23}a_{32}$$M_{21} = a_{12}a_{33} - a_{13}a_{32}, \quad A_{21} = a_{13}a_{32} - a_{12}a_{33}$ ##
By definition of minors and cofactors, we have
Minor of $a_{11} = M_{11} = \begin{vmatrix} a_{22} & a_{23} \\ a_{32} & a_{33} \end{vmatrix} = a_{22} a_{33} - a_{23} a_{32}$
Cofactor of $a_{11} = A_{11} = (-1)^{1+1} M_{11} = a_{22} a_{33} - a_{23} a_{32}$
Minor of $a_{21} = M_{21} = \begin{vmatrix} a_{12} & a_{13} \\ a_{32} & a_{33} \end{vmatrix} = a_{12} a_{33} - a_{13} a_{32}$
Cofactor of $a_{21} = A_{21} = (-1)^{2+1} M_{21} = (-1) (a_{12} a_{33} - a_{13} a_{32}) = - a_{12} a_{33} + a_{13} a_{32}$