Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 3: Q13E (page 165)

Combine the methods of row reduction and cofactor expansion to compute the determinant in Exercise 13.

13. \(\left| {\begin{aligned}{*{20}{c}}{\bf{2}}&{\bf{5}}&{\bf{4}}&{\bf{1}}\\{\bf{4}}&{\bf{7}}&{\bf{6}}&{\bf{2}}\\{\bf{6}}&{ - {\bf{2}}}&{ - {\bf{4}}}&{\bf{0}}\\{ - {\bf{6}}}&{\bf{7}}&{\bf{7}}&{\bf{0}}\end{aligned}} \right|\)

Short Answer

Expert verified

\(\left| {\begin{aligned}{*{20}{c}}2&5&4&1\\4&7&6&2\\6&{ - 2}&{ - 4}&0\\{ - 6}&7&7&0\end{aligned}} \right| = 6\)

Step by step solution

01

Create zero in the fourth column

Add \( - 2\) times row 1 to row 2 to obtain

\(\left| {\begin{aligned}{*{20}{c}}2&5&4&1\\4&7&6&2\\6&{ - 2}&{ - 4}&0\\{ - 6}&7&7&0\end{aligned}} \right| = \left| {\begin{aligned}{*{20}{c}}2&5&4&1\\0&{ - 3}&{ - 2}&0\\6&{ - 2}&{ - 4}&0\\{ - 6}&7&7&0\end{aligned}} \right|\).

02

Use cofactor expansion down the fourth column

\(\begin{aligned}{c}\left| {\begin{aligned}{*{20}{c}}2&5&4&1\\0&{ - 3}&{ - 2}&0\\6&{ - 2}&{ - 4}&0\\{ - 6}&7&7&0\end{aligned}} \right| = - 1\left| {\begin{aligned}{*{20}{c}}0&{ - 3}&{ - 2}\\6&{ - 2}&{ - 4}\\{ - 6}&7&7\end{aligned}} \right| + 0 + 0 + 0\\ = - \left| {\begin{aligned}{*{20}{c}}0&{ - 3}&{ - 2}\\6&{ - 2}&{ - 4}\\{ - 6}&7&7\end{aligned}} \right|\end{aligned}\)

03

Create zero in the first column

Add row 3 to row 2 to obtain

\( - \left| {\begin{aligned}{*{20}{c}}0&{ - 3}&{ - 2}\\6&{ - 2}&{ - 4}\\{ - 6}&7&7\end{aligned}} \right| = - \left| {\begin{aligned}{*{20}{c}}0&{ - 3}&{ - 2}\\0&5&3\\{ - 6}&7&7\end{aligned}} \right|\).

04

Use cofactor expansion down the first column

\(\begin{aligned}{c} - \left| {\begin{aligned}{*{20}{c}}0&{ - 3}&{ - 2}\\0&5&3\\{ - 6}&7&7\end{aligned}} \right| = - \left( {0 + 0 - 6\left| {\begin{aligned}{*{20}{c}}{ - 3}&{ - 2}\\5&3\end{aligned}} \right|} \right)\\ = - \left( { - 6\left( 1 \right)} \right)\\ = 6\end{aligned}\)

Hence, \(\left| {\begin{aligned}{*{20}{c}}2&5&4&1\\4&7&6&2\\6&{ - 2}&{ - 4}&0\\{ - 6}&7&7&0\end{aligned}} \right| = 6\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

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.

\(\left[ {\begin{array}{*{20}{c}}{\bf{3}}&{\bf{2}}\\{\bf{5}}&{\bf{4}}\end{array}} \right],\left[ {\begin{array}{*{20}{c}}{\bf{3}}&{\bf{2}}\\{5 + 3k}&{4 + 2k}\end{array}} \right]\)

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.

\(\left[ {\begin{aligned}{*{20}{c}}a&b\\c&d\end{aligned}} \right],\left[ {\begin{aligned}{*{20}{c}}c&d\\a&b\end{aligned}} \right]\)

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

\(\left| {\begin{array}{*{20}{c}}{\bf{1}}&{\bf{3}}&{ - {\bf{4}}}\\{\bf{2}}&{\bf{0}}&{ - {\bf{3}}}\\{\bf{3}}&{ - {\bf{5}}}&{\bf{2}}\end{array}} \right| = \left| {\begin{array}{*{20}{c}}{\bf{1}}&{\bf{3}}&{ - {\bf{4}}}\\{\bf{0}}&{ - {\bf{6}}}&{\bf{5}}\\{\bf{3}}&{ - {\bf{5}}}&{\bf{2}}\end{array}} \right|\)

Find the determinant in Exercise 15, where \[\left| {\begin{array}{*{20}{c}}{\bf{a}}&{\bf{b}}&{\bf{c}}\\{\bf{d}}&{\bf{e}}&{\bf{f}}\\{\bf{g}}&{\bf{h}}&{\bf{i}}\end{array}} \right| = {\bf{7}}\].

15. \[\left| {\begin{array}{*{20}{c}}{\bf{a}}&{\bf{b}}&{\bf{c}}\\{\bf{d}}&{\bf{e}}&{\bf{f}}\\{{\bf{3g}}}&{{\bf{3h}}}&{{\bf{3i}}}\end{array}} \right|\]

Compute the determinant in Exercise 5 using a cofactor expansion across the first row.

5. \(\left| {\begin{aligned}{*{20}{c}}{\bf{2}}&{\bf{3}}&{ - {\bf{3}}}\\{\bf{4}}&{\bf{0}}&{\bf{3}}\\{\bf{6}}&{\bf{1}}&{\bf{5}}\end{aligned}} \right|\)
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free