Find the minors and cofactors of the elements $a_{11}$ and $a_{21}$ in the determinant $\Delta = \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix}$.
By the definition of minors and cofactors,we have:
$1$. Minor of $a_{11}$ $(M_{11})$: Delete the first row and first column.
$M_{11} = \begin{vmatrix} a_{22} & a_{23} \\ a_{32} & a_{33} \end{vmatrix} = a_{22}a_{33} - a_{23}a_{32}$
$2$. Cofactor of $a_{11}$ $(A_{11})$:
$A_{11} = (-1)^{1+1} M_{11} = 1 \times (a_{22}a_{33} - a_{23}a_{32}) = a_{22}a_{33} - a_{23}a_{32}$
$3$. Minor of $a_{21}$ $(M_{21})$: Delete the second row and first column.
$M_{21} = \begin{vmatrix} a_{12} & a_{13} \\ a_{32} & a_{33} \end{vmatrix} = a_{12}a_{33} - a_{13}a_{32}$
$4$. Cofactor of $a_{21}$ $(A_{21})$:
$A_{21} = (-1)^{2+1} M_{21} = -1 \times (a_{12}a_{33} - a_{13}a_{32}) = -a_{12}a_{33} + a_{13}a_{32}$
$1$. Minor of $a_{11}$ $(M_{11})$: Delete the first row and first column.
$M_{11} = \begin{vmatrix} a_{22} & a_{23} \\ a_{32} & a_{33} \end{vmatrix} = a_{22}a_{33} - a_{23}a_{32}$
$2$. Cofactor of $a_{11}$ $(A_{11})$:
$A_{11} = (-1)^{1+1} M_{11} = 1 \times (a_{22}a_{33} - a_{23}a_{32}) = a_{22}a_{33} - a_{23}a_{32}$
$3$. Minor of $a_{21}$ $(M_{21})$: Delete the second row and first column.
$M_{21} = \begin{vmatrix} a_{12} & a_{13} \\ a_{32} & a_{33} \end{vmatrix} = a_{12}a_{33} - a_{13}a_{32}$
$4$. Cofactor of $a_{21}$ $(A_{21})$:
$A_{21} = (-1)^{2+1} M_{21} = -1 \times (a_{12}a_{33} - a_{13}a_{32}) = -a_{12}a_{33} + a_{13}a_{32}$
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