Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 3: Q22E (page 165)

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.

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

Short Answer

Expert verified

The row operation replaces row 2 by k times row 1 plus row 2, and the determinant remains unchanged.

Step by step solution

01

Find the determinant of the first matrix

The determinant of the matrix \(\left[ {\begin{array}{*{20}{c}}3&2\\5&4\end{array}} \right]\) can be calculated as shown below:

\(\begin{array}{c}\left| {\begin{array}{*{20}{c}}3&2\\5&4\end{array}} \right| = 12 - 10\\ = 2\end{array}\)

02

Find the determinant of the second matrix

The determinant of the matrix \(\left[ {\begin{array}{*{20}{c}}a&b\\{kc}&{kd}\end{array}} \right]\) can be calculated as shown below:

\(\begin{array}{c}\left| {\begin{array}{*{20}{c}}3&2\\{5 + 3k}&{4 + 2k}\end{array}} \right| = 3\left( {4 + 2k} \right) - 2\left( {5 + 3k} \right)\\ = 12 + 6k - 10 - 6k\\ = 2\end{array}\)

So, the row operation replaces row 2 by k times row 1 plus row 2, and the determinant remains unchanged.

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

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

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

In Exercises 39 and 40, \(A\) is an \(n \times n\) matrix. Mark each statement True or False. Justify each answer.

39.

a. An \(n \times n\) determinant is defined by determinants of \(\left( {n - 1} \right) \times \left( {n - 1} \right)\) submatrices.

b. The \(\left( {i,j} \right)\)-cofactor of a matrix \(A\) is the matrix \({A_{ij}}\) obtained by deleting from A its \(i{\mathop{\rm th}\nolimits} \) row and \[j{\mathop{\rm th}\nolimits} \]column.

Compute the determinant in Exercise 4 using a cofactor expansion across the first row. Also compute the determinant by a cofactor expansion down the second column.

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

Compute the determinants of the elementary matrices given in Exercise 25-30.

30. \(\left[ {\begin{aligned}{*{20}{c}}0&1&0\\1&0&0\\0&0&1\end{aligned}} \right]\).

Combine the methods of row reduction and cofactor expansion to compute the determinant in Exercise 12.

12. \(\left| {\begin{aligned}{*{20}{c}}{ - {\bf{1}}}&{\bf{2}}&{\bf{3}}&{\bf{0}}\\{\bf{3}}&{\bf{4}}&{\bf{3}}&{\bf{0}}\\{{\bf{11}}}&{\bf{4}}&{\bf{6}}&{\bf{6}}\\{\bf{4}}&{\bf{2}}&{\bf{4}}&{\bf{3}}\end{aligned}} \right|\)
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

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