Chapter 3.2, Problem 43E

Verify that det A = det B + det C, where

A = $\left[\begin{array}{ccc}{\text{a}}_{11}& {\text{a}}_{12}& {\text{u}}_1+{\text{v}}_1\\ {\text{a}}_{21}& {\text{a}}_{22}& {\text{u}}_2+{\text{v}}_2\\ {\text{a}}_{31}& {\text{a}}_{32}& {\text{u}}_3+{\text{v}}_3\end{array}\right]$, B = $\left[\begin{array}{ccc}{\text{a}}_{11}& {\text{a}}_{12}& {\text{u}}_1\\ {\text{a}}_{21}& {\text{a}}_{22}& {\text{u}}_2\\ {\text{a}}_{31}& {\text{a}}_{32}& {\text{u}}_3\end{array}\right]$, C = $\left[\begin{array}{ccc}{\text{a}}_{11}& {\text{a}}_{12}& {\text{v}}_1\\ {\text{a}}_{21}& {\text{a}}_{22}& {\text{v}}_2\\ {\text{a}}_{31}& {\text{a}}_{32}& {\text{v}}_3\end{array}\right]$

Note, however that, A is not the same as B +C.