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Find minors and cofactors of the elements $a_{11}, a_{21}$ in the determinant
$\Delta=\left|\begin{array}{lll}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{array}\right|$
Solution
Solution By definition of minors and cofactors, we have
Minor of $a_{11}=\mathrm{M}_{11}=\left|\begin{array}{ll}a_{22} & a_{23} \\ a_{32} & a_{33}\end{array}\right|=a_{22} a_{33}-a_{23} a_{32}$
Cofactor of $a_{11}=\mathrm{A}_{11}=(-1)^{1+1} \quad \mathrm{M}_{11}=a_{22} a_{33}-a_{23} a_{32}$
Minor of $a_{21}=\mathrm{M}_{21}=\left|\begin{array}{ll}a_{12} & a_{13} \\ a_{32} & a_{33}\end{array}\right|=a_{12} a_{33}-a_{13} a_{32}$
Cofactor of $a_{21}=\mathrm{A}_{21}=(-1)^{2+1} \quad \mathrm{M}_{21}=(-1)\left(a_{12} a_{33}-a_{13} a_{32}\right)=-a_{12} a_{33}+a_{13} a_{32}$
