Question

Find minors and cofactors of all the element of the determinant $\left|\begin{array}{cc}1& -2\\ 4& 3\end{array}\right|$

Answer 1

Given: $\Delta =\left|\begin{array}{cc}1& -2\\ 4& 3\end{array}\right|$
Minor of $a_{11}=M_{11}=3$
Minor of $a_{12}=M_{12}=4$
Minor of $a_{21}=M_{21}=-2$
Minor of $a_{22}=M_{22}=1$
Cofactor of $a_{11}=A_{11}=(-1{)}^{1+1}{M}_{11}$
$\Rightarrow A_{11}=(-1{)}^2\cdot 3=3$
Cofactor of $a_{12}=A_{12}=(-1{)}^{1+2}{M}_{12}$
$\Rightarrow A_{12}=(-1{)}^3\cdot 4=(-1\left)\right(4)=-4$
Cofactor of $a_{21}=(-1{)}^{2+1}{M}_{21}$
$\Rightarrow A_{21}=(-1{)}^3(-2)=(-1\left)\right(-2)=2$
Cofactor of $a_{22}=A_{22}=(-1{)}^{2+2}\cdot M_{22}$
$\Rightarrow A_{22}=(-1{)}^4\cdot (1)=(1\left)\right(1)=1$
Therefore,
$M_{11}=3,M_{12}=4,M_{21}=-2,M_{22}=1$ and
$A_{11}=3,A_{12}=-4,A_{21}=2,A_{22}=1$