Question

Using cofactors of elements of second row, evaluate $\Delta =\left|\begin{array}{ccc}5& 3& 8\\ 2& 0& 1\\ 1& 2& 3\end{array}\right|$

Answer 1

$△=\left|\begin{array}{ccc}5& 3& 8\\ 2& 0& 1\\ 1& 2& 3\end{array}\right|$

Cofactors of elements of second row are

$C_{21}=(-1{)}^{2+1}\left|\begin{array}{cc}3& 8\\ 2& 3\end{array}\right|$

$\Rightarrow C_{21}=-(-7)=7$

$C_{22}=(-1{)}^{2+2}\left|\begin{array}{cc}5& 8\\ 1& 3\end{array}\right|$

$\Rightarrow C_{22}=15-8=7$

$C_{23}=(-1{)}^{2+3}\left|\begin{array}{cc}5& 3\\ 1& 2\end{array}\right|$

$\Rightarrow C_{23}=10-3=7$

Now, $\Delta =2C_{21}+0C_{22}+1C_{23}$

$\Delta =2\left(7\right)+0\left(7\right)+1\left(7\right)=21$