Question

Find minors and cofactors of the elements $a_{11},a_{21}$ in the determinant
$\Delta =\left|\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right|$

Answer 1

Given : $\Delta =\left|\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right|$
Minor of $a_{11}=M_{11}=\left|\begin{array}{cc}{a}_{22}& a_{23}\\ a_{32}& a_{33}\end{array}\right|=a_{22}{a}_{33}-a_{23}{a}_{32}$
Minor of $a_{21}=M_{21}=\left|\begin{array}{cc}{a}_{12}& a_{13}\\ a_{32}& a_{33}\end{array}\right|=a_{12}{a}_{33}-a_{32}{a}_{13}$
Cofactor of $a_{11}=A_{11}=(-1{)}^{1+1}.M_{11}=(-1{)}^2.(a_{22}{a}_{33}-a_{23}{a}_{32})$
$\Rightarrow A_{11}=a_{22}{a}_{33}-a_{23}{a}_{32}$
Cofactor of $a_{21}=A_{21}=(-1{)}^{2+1}\cdot M_{21}=(-1{)}^3.(a_{12}{a}_{33}-a_{32}{a}_{13})$
$\Rightarrow A_{21}=(-1)\cdot (a_{12}{a}_{33}-a_{32}{a}_{13})$
$\Rightarrow A_{21}=a_{32}{a}_{13}-a_{12}{a}_{33}$