The sum of the cofactors of the elements of second row of the matrix $\left[\begin{array}{ccc}1 & 3 & 2 \\…
The sum of the cofactors of the elements of second row of the matrix $\left[\begin{array}{ccc}1 & 3 & 2 \\ -2 & 0 & 1 \\ 5 & 2 &1\end{array}\right]$ is
23
3
5
-23
Solution
Co-factors of second row of matrix are
$\begin{array}{l}
A_{21}=(-1)^{2+1}\left|\begin{array}{ll}
3 & 2 \\
2 & 1
\end{array}\right|=(-1)^{3}(-1)=1 \\
A_{22}=(-1)^{2+2}\left|\begin{array}{ll}
1 & 2 \\
5 & 1
\end{array}\right|=(-1)^{4}(-9)=-9 \\
A_{23}=(-1)^{2+3}\left[\begin{array}{ll}
1 & 3 \\
5 & 2
\end{array}\right]=(-1)^{5}(-13)=13 \\
\text { Their sum }=A_{21}+A_{12}+A_{23}=1-9+13=5
\end{array}$