Question

Write minors and cofactors of elements of following determinants.

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

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

Answer 1

Here, $\left|\begin{array}{cc}2& -4\\ 0& 3\end{array}\right|\therefore$ Minors, $M_{11}=3,M_{12}=0,M_{21}=-4and{M}_{22}=2$

Also cofactors, $A_{11}=(-1{)}^{1+1}{M}_{11}=1\times 3=3A_{12}=(-1{)}^{1+2}{M}_{12}=(-1)\times 0=0A_{21}=(-1{)}^{2+1}{M}_{21}=(-1)\times (-4)=4$
and $A_{22}=(-1{)}^{2+2}{M}_{22}=1\times 2=2$

Here, $\left|\begin{array}{cc}a& c\\ b& d\end{array}\right|\therefore$ Minors, $M_{11}=d,M_{12}=b,M_{21}=cand{M}_{22}=a$

Also cofactors, $A_{11}=(-1{)}^{1+1}{M}_{11}=1\times d=d{A}_{12}=(-1{)}^{1+2}{M}_{12}=(-1)\times b=-b{A}_{21}=(-1{)}^{2+1}{M}_{21}=(-1)\times c=-c{A}_{22}=(-1{)}^{2+2}{M}_{22}=1\times a=a$