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Chapter 3: Q41E (page 165)

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

Expert verified

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

Step by step solution

01

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

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

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

02

conclusion

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

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Most popular questions from this chapter

Compute the determinant in Exercise 1 using a cofactor expansion across the first row. Also compute the determinant by a cofactor expansion down the second column.

  1. \(\left| {\begin{aligned}{*{20}{c}}{\bf{3}}&{\bf{0}}&{\bf{4}}\\{\bf{2}}&{\bf{3}}&{\bf{2}}\\{\bf{0}}&{\bf{5}}&{ - {\bf{1}}}\end{aligned}} \right|\)

Compute the determinant in Exercise 2 using a cofactor expansion across the first row. Also compute the determinant by a cofactor expansion down the second column.

  1. \(\left| {\begin{aligned}{*{20}{c}}{\bf{0}}&{\bf{4}}&{\bf{1}}\\{\bf{5}}&{ - {\bf{3}}}&{\bf{0}}\\{\bf{2}}&{\bf{3}}&{\bf{1}}\end{aligned}} \right|\)

Question: In Exercise 11, compute the adjugate of the given matrix, and then use Theorem 8 to give the inverse of the matrix.

11. \(\left( {\begin{array}{*{20}{c}}{\bf{0}}&{ - {\bf{2}}}&{ - {\bf{1}}}\\{\bf{5}}&{\bf{0}}&{\bf{0}}\\{ - {\bf{1}}}&{\bf{1}}&{\bf{1}}\end{array}} \right)\)

Let \(A = \left[ {\begin{aligned}{*{20}{c}}a&b\\c&d\end{aligned}} \right]\) and let \(k\) be a scalar. Find a formula that relates \(\det kA\) to \(k\) and \(\det A\).

Let \(u = \left[ {\begin{aligned}{*{20}{c}}a\\b\end{aligned}} \right]\), and \(v = \left[ {\begin{aligned}{*{20}{c}}c\\{\bf{0}}\end{aligned}} \right]\), where a, b, and c are positive integers (for simplicity). Compute the area of the parallelogram determined by u, v, \({\bf{u}} + {\bf{v}}\), and 0, and compute the determinant of \(\left[ {\begin{aligned}{*{20}{c}}{\bf{u}}&{\bf{v}}\end{aligned}} \right]\), and \[\left[ {\begin{aligned}{*{20}{c}}{\bf{v}}&{\bf{u}}\end{aligned}} \right]\]. Draw a picture and explain what you find.
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