Question

Write minors and cofactors of elements of following determinants

$\left|\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right|$

$\left|\begin{array}{ccc}1& 0& 4\\ 3& 5& -1\\ 0& 1& 2\end{array}\right|$

Answer 1

Here,

$\left|\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right|$
Minors of elements of first row are
$M_{11}=\left|\begin{array}{cc}1& 0\\ 0& 1\end{array}\right|=1-0=1,M_{12}=\left|\begin{array}{cc}0& 0\\ 0& 1\end{array}\right|=0-0=0$

and $M_{13}=\left|\begin{array}{cc}0& 1\\ 0& 0\end{array}\right|=0-0=0$
Minors of elements of second row are
$M_{21}=\left|\begin{array}{cc}0& 0\\ 0& 1\end{array}\right|=0-0=0M_{22}=\left|\begin{array}{cc}1& 0\\ 0& 1\end{array}\right|=1-0=1and{M}_{23}=\left|\begin{array}{cc}1& 0\\ 0& 0\end{array}\right|=0-0=0$
Minors of elements of third row are
$M_{31}=\left|\begin{array}{cc}0& 0\\ 1& 0\end{array}\right|=0-0=0M_{32}=\left|\begin{array}{cc}1& 0\\ 0& 0\end{array}\right|=0-0=0and{M}_{33}=\left|\begin{array}{cc}1& 0\\ 0& 1\end{array}\right|=1-0=1$

Hence, cofactors of elements of first row are
$A_{11}=(-1{)}^{1+1}{M}_{11}=1\times 1=1,A_{12}=(-1{)}^{1+2}{M}_{12}=-1\times 0=0A_{13}=(-1{)}^{1+3}{M}_{13}=1\times 0=0$

cofactors of elements of second row are
$A_{21}=(-1{)}^{2+1}{M}_{21}=1\times 0=0,A_{22}=(-1{)}^{2+2}{M}_{22}=1\times 1=1A_{23}=(-1{)}^{2+3}{M}_{23}=-1\times 0=0$

Coafactors of elements of third row are
$A_{31}=(-1{)}^{3+1}{M}_{31}=1\times 0=0,A_{32}=(-1{)}^{3+2}{M}_{32}=-1\times 0=0A_{33}=(-1{)}^{3+3}{M}_{33}=1\times 1=1$

Here,
$\left|\begin{array}{ccc}1& 0& 4\\ 3& 5& -1\\ 0& 1& 2\end{array}\right|$
Minors of elements of first row are
$M_{11}=\left|\begin{array}{cc}5& -1\\ 1& 2\end{array}\right|=10+1=11,M_{12}=\left|\begin{array}{cc}3& -1\\ 0& 2\end{array}\right|=6-0=6and{M}_{13}=\left|\begin{array}{cc}3& 5\\ 0& 1\end{array}\right|=3-0=3$
Minors of elements of second row are
$M_{21}=\left|\begin{array}{cc}0& 4\\ 1& 2\end{array}\right|=0-4=-4M_{22}=\left|\begin{array}{cc}1& 4\\ 0& 2\end{array}\right|=2-0=2and{M}_{23}=\left|\begin{array}{cc}1& 0\\ 0& 1\end{array}\right|=1-0=1$
Minors of elements of third row are
$M_{31}=\left|\begin{array}{cc}0& 4\\ 5& -1\end{array}\right|=0-20=-20M_{32}=\left|\begin{array}{cc}1& 4\\ 3& -1\end{array}\right|=-1-12=-13and{M}_{33}=\left|\begin{array}{cc}1& 0\\ 3& 5\end{array}\right|=5-0=5$

Hence, cofactors of elements of first row are
$A_{11}=(-1{)}^{1+1}{M}_{11}=1\times 11=11,A_{12}=(-1{)}^{1+2}{M}_{12}=-1\times 6=-6A_{13}=(-1{)}^{1+3}{M}_{13}=1\times 3=3$
cofactors of elements of second row are
$A_{21}=(-1{)}^{2+1}{M}_{21}=-1\times -4=4,A_{22}=(-1{)}^{2+2}{M}_{22}=1\times 2=2A_{23}=(-1{)}^{2+3}{M}_{23}=1\times 1=-1$
Coafactors of elements of third row are
$A_{31}=(-1{)}^{3+1}{M}_{31}=1\times -20=-20,A_{32}=(-1{)}^{3+2}{M}_{32}=-1\times -13=13A_{33}=(-1{)}^{3+3}{M}_{33}=1\times 5=5$