/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Q9E Compute the determinant in Exerc... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Q9E (page 165)

Compute the determinant in Exercise 9 by cofactor expansions. At each step, choose a row or column that involves the least amount of computation.

9. \(\left| {\begin{array}{*{20}{c}}{\bf{4}}&{\bf{0}}&{\bf{0}}&{\bf{5}}\\{\bf{1}}&{\bf{7}}&{\bf{2}}&{ - {\bf{5}}}\\{\bf{3}}&{\bf{0}}&{\bf{0}}&{\bf{0}}\\{\bf{8}}&{\bf{3}}&{\bf{1}}&{\bf{7}}\end{array}} \right|\)

Short Answer

Expert verified

\(\left| {\begin{array}{*{20}{c}}4&0&0&5\\1&7&2&{ - 5}\\3&0&0&0\\8&3&1&7\end{array}} \right| = 15\)

Step by step solution

01

Write the determinant formula

The determinant computed by cofactor expansion across the ith row is

\(\det A = {a_{i1}}{C_{i1}} + {a_{i2}}{C_{i2}} + \cdots + {a_{in}}{C_{in}}\).

Here, A is an \(n \times n\) matrix, and \({C_{ij}} = {\left( { - 1} \right)^{i + j}}{A_{ij}}\).

For minimum computation, choose a row or column that has zero as maximum entries.

02

Use cofactor expansion across the third row

\(\begin{array}{c}\left| {\begin{array}{*{20}{c}}4&0&0&5\\1&7&2&{ - 5}\\3&0&0&0\\8&3&1&7\end{array}} \right| = 3\left| {\begin{array}{*{20}{c}}0&0&5\\7&2&{ - 5}\\3&1&7\end{array}} \right| + 0 + 0 + 0\\ = 3\left| {\begin{array}{*{20}{c}}0&0&5\\7&2&{ - 5}\\3&1&7\end{array}} \right|\end{array}\)

03

Use cofactor expansion across the first row

\(\begin{array}{c}\left| {\begin{array}{*{20}{c}}4&0&0&5\\1&7&2&{ - 5}\\3&0&0&0\\8&3&1&7\end{array}} \right| = 3\left( {0 + 0 + 5\left| {\begin{array}{*{20}{c}}7&2\\3&1\end{array}} \right|} \right)\\ = 3\left( {5\left( 1 \right)} \right)\\ = 15\end{array}\)

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.

Over 30 million students worldwide already upgrade their learning with 91影视!

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

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?

Question: Use Cramer鈥檚 rule to compute the solutions of the systems in Exercises1-6.

2. \(\begin{array}{l}4{x_1} + {x_2} = 6\\3{x_1} + 2{x_2} = 7\end{array}\)

Construct a random \({\bf{4}} \times {\bf{4}}\) matrix A with integer entries between \( - {\bf{9}}\) and 9, and compare det A with det\({A^T}\), \(det\left( { - A} \right)\), \(det\left( {{\bf{2}}A} \right)\), and \(det\left( {{\bf{10}}A} \right)\). Repeat with two other random \({\bf{4}} \times {\bf{4}}\) integer matrices, and make conjectures about how these determinants are related. (Refer to Exercise 36 in Section 2.1.) Then check your conjectures with several random \({\bf{5}} \times {\bf{5}}\) and \({\bf{6}} \times {\bf{6}}\) integer matrices. Modify your conjectures, if necessary, and report your results.

Compute the determinant in Exercise 6 using a cofactor expansion across the first row.

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

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

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Probability and Statistics

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.