Question

Write minors and cofactors of the elements of the following determinants:
$\left(i\right)\left|\begin{array}{cc}2& -4\\ 0& 3\end{array}\right|$
$\left(ii\right)\left|\begin{array}{cc}a& c\\ b& d\end{array}\right|$
$\left(iii\right)\left|\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right|$
$\left(iv\right)\left|\begin{array}{ccc}1& 0& 4\\ 3& 5& -1\\ 0& 1& 2\end{array}\right|$

Answer 1

$\left(i\right)\left|\begin{array}{cc}2& -4\\ 0& 3\end{array}\right|Minor\Rightarrow (6-0)=6cofactors\left|\begin{array}{cc}3& 0\\ -4& 2\end{array}\right|\left(ii\right)\left|\begin{array}{cc}a& c\\ b& d\end{array}\right|Minor\Rightarrow (ad-bc)=6cofactors\left|\begin{array}{cc}d& c\\ c& a\end{array}\right|\left(iii\right)\left|\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right|Minor=\left|\begin{array}{cc}1& 0\\ 0& 1\end{array}\right|\Rightarrow 1cofactors=\left|\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right|\left(iv\right)\left|\begin{array}{ccc}1& 0& 4\\ 3& 5& -1\\ 0& 1& 2\end{array}\right|Minor=\left|\begin{array}{cc}5& -1\\ 1& 2\end{array}\right|=10+1=11cofactors=\left|\begin{array}{ccc}11& -6& 3\\ 4& 2& -1\\ -20& 13& 5\end{array}\right|$