Answer
See solution
Work Step by Step
Computing det A by the cofactor expansion down column 3 is equal to computing the sum of the determinants of B and C by the cofactor expansion down column 3
$det A=(u_1+v_1)\left| \begin{bmatrix}
a_{21}&a_{22}\\a_{31}&a_{32}
\end{bmatrix}\right|-(u_2+v_2)\left| \begin{bmatrix}
a_{11}&a_{12}\\a_{31}&a_{32}
\end{bmatrix}\right|+(u_3+v_3)\left| \begin{bmatrix}
a_{11}&a_{12}\\a_{21}&a_{22}
\end{bmatrix}\right|=(u_1)\left| \begin{bmatrix}
a_{21}&a_{22}\\a_{31}&a_{32}
\end{bmatrix}\right|-(u_2)\left| \begin{bmatrix}
a_{11}&a_{12}\\a_{31}&a_{32}
\end{bmatrix}\right|+(u_3)\left| \begin{bmatrix}
a_{11}&a_{12}\\a_{21}&a_{22}
\end{bmatrix}\right|+(v_1)\left| \begin{bmatrix}
a_{21}&a_{22}\\a_{31}&a_{32}
\end{bmatrix}\right|-(v_2)\left| \begin{bmatrix}
a_{11}&a_{12}\\a_{31}&a_{32}
\end{bmatrix}\right|+(v_3)\left| \begin{bmatrix}
a_{11}&a_{12}\\a_{21}&a_{22}
\end{bmatrix}\right|=det(B)+det(C)$
