Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 3: Q19E (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.

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

Short Answer

Expert verified

The row operation swaps row 1 and 2 of the matrix and reverses the sign of the determinant.

Step by step solution

01

Find the determinant of the first matrix

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

\(\left| {\begin{aligned}{*{20}{c}}a&b\\c&d\end{aligned}} \right| = ad - bc\)

02

Find the determinant of the second matrix

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

\(\begin{aligned}{c}\left| {\begin{aligned}{*{20}{c}}c&d\\a&b\end{aligned}} \right| = bc - ad\\ = - \left( {ad - bd} \right)\end{aligned}\)

So, the row operation swaps row 1 and 2 of the matrix and reverses the sign of the determinant.

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

Question: In Exercises 31–36, mention an appropriate theorem in your explanation.

34. Let A and P be square matrices, with P invertible. Show that \(det\left( {PA{P^{ - {\bf{1}}}}} \right) = det{\rm{ }}A\).

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

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

In Exercises 24–26, use determinants to decide if the set of vectors is linearly independent.

25. \(\left( {\begin{aligned}{*{20}{c}}7\\{ - 4}\\{ - 6}\end{aligned}} \right)\), \(\left( {\begin{aligned}{*{20}{c}}{ - {\bf{8}}}\\{\bf{5}}\\7\end{aligned}} \right)\), \(\left( {\begin{aligned}{*{20}{c}}{\bf{7}}\\{\bf{0}}\\{ - {\bf{5}}}\end{aligned}} \right)\)

The expansion of a \({\bf{3}} \times {\bf{3}}\) determinant can be remembered by the following device. Write a second type of the first two columns to the right of the matrix, and compute the determinant by multiplying entries on six diagonals.

Add the downward diagonal products and subtract the upward products. Use this method to compute the determinants in Exercises 15-18. Warning: This trick does not generalize in any reasonable way to \({\bf{4}} \times {\bf{4}}\) or larger matrices.

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

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

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

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