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Q17E

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Linear Algebra and its Applications
Found in: Page 165
Linear Algebra and its Applications

Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

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Short Answer

Find the determinant in Exercise 17, where \[\left| {\begin{aligned}{*{20}{c}}{\bf{a}}&{\bf{b}}&{\bf{c}}\\{\bf{d}}&{\bf{e}}&{\bf{f}}\\{\bf{g}}&{\bf{h}}&{\bf{i}}\end{aligned}} \right| = {\bf{7}}\].

17. \[\left| {\begin{aligned}{*{20}{c}}{{\bf{a}} + {\bf{d}}}&{{\bf{b}} + {\bf{e}}}&{{\bf{c}} + {\bf{f}}}\\{\bf{d}}&{\bf{e}}&{\bf{f}}\\{\bf{g}}&{\bf{h}}&{\bf{i}}\end{aligned}} \right|\]

Hence, \[\left| {\begin{aligned}{*{20}{c}}{a + d}&{b + e}&{c + f}\\d&e&f\\g&h&i\end{aligned}} \right| = 7\].

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Reduce the given determinant

At row 1, add \[ - 1\] time row 2 to row 1 to obtain:

\[\left| {\begin{aligned}{*{20}{c}}{a + d}&{b + e}&{c + f}\\d&e&f\\g&h&i\end{aligned}} \right| = \left| {\begin{aligned}{*{20}{c}}a&b&c\\d&e&f\\g&h&i\end{aligned}} \right|\]

Step 2: Use the given statement

The statement is \[\left| {\begin{aligned}{*{20}{c}}a&b&c\\d&e&f\\g&h&i\end{aligned}} \right| = 7\].

Step 3: Conclusion

Therefore,

\[\left| {\begin{aligned}{*{20}{c}}{a + d}&{b + e}&{c + f}\\d&e&f\\g&h&i\end{aligned}} \right| = 7\]

Most popular questions for Math Textbooks

Question: 17. Show that if A is \({\bf{2}} \times {\bf{2}}\), then Theorem 8 gives the same formula for \({A^{ - {\bf{1}}}}\) as that given by theorem 4 in Section 2.2.

Each equation in Exercises 1-4 illustrates a property of determinants. State the property

\(\left| {\begin{array}{*{20}{c}}{\bf{3}}&{ - {\bf{6}}}&{\bf{9}}\\{\bf{3}}&{\bf{5}}&{ - {\bf{5}}}\\{\bf{1}}&{\bf{3}}&{\bf{3}}\end{array}} \right| = {\bf{3}}\left| {\begin{array}{*{20}{c}}{\bf{1}}&{ - {\bf{2}}}&{\bf{3}}\\{\bf{3}}&{\bf{5}}&{ - {\bf{5}}}\\{\bf{1}}&{\bf{3}}&{\bf{3}}\end{array}} \right|\)

Let \(u = \left[ {\begin{array}{*{20}{c}}3\\0\end{array}} \right]\), and \(v = \left[ {\begin{array}{*{20}{c}}1\\2\end{array}} \right]\). Compute the area of the parallelogram

determined by u, v, \({\bf{u}} + {\bf{v}}\), and 0, and compute the determinant of \(\left[ {\begin{array}{*{20}{c}}{\bf{u}}&{\bf{v}}\end{array}} \right]\). How do they compare? Replace the first entry of v by an arbitrary number x, and repeat the problem. Draw a picture and explain what you find.

Use Exercise 25-28 to answer the questions in Exercises 31 ad 32. Give reasons for your answers.

31. What is the determinant of an elementary row replacement matrix?

In Exercise 19-24, explore the effect of an elementary row operation on the determinant of a matrix. In each case, state the row operation and describe how it affects the determinant.

\(\left[ {\begin{array}{*{20}{c}}a&b\\c&d\end{array}} \right],\left[ {\begin{array}{*{20}{c}}{a + kc}&{b + kd}\\c&d\end{array}} \right]\)
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