Question

Find the Minors and Cofactors of the elements of the following determinants:

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

Answer 1

(i) $A=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

The elements of the minor of the matrix $A$ will be given by

$M_{1,1}=\left|\begin{array}{cc}1& 0\\ 0& 1\end{array}\right|=1j$

$M_{1,2}=\left|\begin{array}{cc}0& 0\\ 0& 1\end{array}\right|=0$

$M_{1,3}=\left|\begin{array}{cc}0& 1\\ 0& 0\end{array}\right|=0$

$M_{2,1}=\left|\begin{array}{cc}0& 0\\ 0& 1\end{array}\right|=0$

$M_{2,2}=\left|\begin{array}{cc}1& 0\\ 0& 1\end{array}\right|=1$

$M_{2,3}=\left|\begin{array}{cc}1& 0\\ 0& 0\end{array}\right|=0$

$M_{3,1}=\left|\begin{array}{cc}0& 0\\ 1& 0\end{array}\right|=0$

$M_{3,2}=\left|\begin{array}{cc}1& 0\\ 0& 0\end{array}\right|=0$

$M_{3,3}=\left|\begin{array}{cc}1& 0\\ 0& 1\end{array}\right|=1$

Hence, $M=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

The cofactor matrix will be given by $C_{i,j}=(-1{)}^{i+j}{M}_{i,j}$

$\therefore C=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

(ii) $B=\left[\begin{array}{ccc}1& 0& 4\\ 3& 5& -2\\ 0& 1& 2\end{array}\right]$

$M_{1,1}=\left|\begin{array}{cc}5& -2\\ 1& 2\end{array}\right|=12$

$M_{1,2}=\left|\begin{array}{cc}3& -2\\ 0& 2\end{array}\right|=6$

$M_{1,3}=\left|\begin{array}{cc}3& 5\\ 0& 1\end{array}\right|=3$

$M_{2,1}=\left|\begin{array}{cc}0& 4\\ 1& 2\end{array}\right|=-4$

$M_{2,2}=\left|\begin{array}{cc}1& 4\\ 0& 2\end{array}\right|=2$

$M_{2,3}=\left|\begin{array}{cc}1& 0\\ 0& 1\end{array}\right|=1$

$M_{3,1}=\left|\begin{array}{cc}0& 4\\ 5& -2\end{array}\right|=-20$

$M_{3,2}=\left|\begin{array}{cc}1& 4\\ 3& -2\end{array}\right|=-14$

$M_{3,3}=\left|\begin{array}{cc}1& 0\\ 3& 5\end{array}\right|=5$

Hence, $M=\left[\begin{array}{ccc}12& 6& 3\\ -4& 2& 1\\ -20& -14& 5\end{array}\right]$

The cofactor matrix will be given by $C_{i,j}=(-1{)}^{i+j}{M}_{i,j}$

$\therefore C=\left[\begin{array}{ccc}12& -6& 3\\ 4& 2& -1\\ -20& 14& 5\end{array}\right]$