Question

Question 41: Verify that \(\det A = \det B + \det C\), where \(A = \left( {\begin{aligned}{{}}{a + e}&{b + f}\\c&d\end{aligned}} \right)\), \(B = \left( {\begin{aligned}{{}}a&b\\c&d\end{aligned}} \right)\), \(C = \left( {\begin{aligned}{{}}e&f\\c&d\end{aligned}} \right)\).

Short answer

It is verified that \(\det A = \det B + \det C\).

Step-by-step solution

Verify that \(\det A = \det B + \det C\)

The determinant of matrix \(A\)is calculated below:

\(\begin{aligned}{}\det A &= \left| {\begin{aligned}{{}}{a + e}&{b + f}\\c&d\end{aligned}} \right|\\ &= \left( {a + e} \right)d - c\left( {b + f} \right)\\ &= ad + ed - bc - cf\\ &= \left( {ad - bc} \right) + \left( {ed - cf} \right)\\ &= \det B + \det C\end{aligned}\)

conclusion

Thus, it is verified that \(\det A = \det B + \det C\).