Question

Write minors and cofactors of the elements of following determinants
(i)$\left|\begin{array}{cc}2& -4\\ 0& 3\end{array}\right|$
(ii)$\left|\begin{array}{cc}a& c\\ b& d\end{array}\right|$

Answer 1

(i) $\left|\begin{array}{cc}2& -4\\ 0& 3\end{array}\right|$

Minors:

$M_{11}=3$ ; $M_{12}=0$

$M_{21}=-4$ ; $M_{22}=2$

Cofactors:

$C_{11}=(-1{)}^{1+1}{M}_{11}$

$\Rightarrow C_{11}=3$

$C_{12}=(-1{)}^{1+2}{M}_{12}$

$\Rightarrow C_{12}=0$

$C_{21}=(-1{)}^{2+1}{M}_{21}$

$\Rightarrow C_{21}=-(-4)=4$

$C_{22}=(-1{)}^{2+2}{M}_{22}$

$\Rightarrow C_{22}=2$

(ii) $\left|\begin{array}{cc}a& c\\ b& d\end{array}\right|$

Minors:

$M_{11}=d$ ; $M_{12}=b$

$M_{21}=c$ ; $M_{22}=a$

Cofactors:

$C_{11}=(-1{)}^{1+1}{M}_{11}$

$\Rightarrow C_{11}=d$

$C_{12}=(-1{)}^{1+2}{M}_{12}$

$\Rightarrow C_{12}=-b$

$C_{21}=(-1{)}^{2+1}{M}_{21}$

$\Rightarrow C_{21}=-c$

$C_{22}=(-1{)}^{2+2}{M}_{22}$

$\Rightarrow C_{22}=a$