Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 3: Q30Q (page 165)

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

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

Short Answer

Expert verified

The determinant of the matrix is \( - 1\).

Step by step solution

01

Compute the determinant of the elementary matrices

Theorem 1states that the determinant of an \(n \times n\) matrix A can be computed by cofactor expansion across any row or down any column. Expansion across the

\(i{\mathop{\rm th}\nolimits} \)row using cofactors in \({C_{ij}} = {\left( { - 1} \right)^{i + j}}\det \,{A_{ij}}\) gives \(\det \,A = {a_{i1}}{C_{i1}} + {a_{i2}}{C_{i2}} + ... + {a_{in}}{C_{in}}\).

Cofactor expansion across row 1 is shown below:

\[\begin{aligned}{c}\left| {\begin{aligned}{*{20}{c}}0&1&0\\1&0&0\\0&0&1\end{aligned}} \right| = - 1\left| {\begin{aligned}{*{20}{c}}1&0\\0&1\end{aligned}} \right|\\ = - 1\left( 1 \right)\\ = - 1\end{aligned}\]

Thus, the determinant of the matrix is \( - 1\).

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

Find the determinants in Exercises 5-10 by row reduction to echelon form.

\(\left| {\begin{aligned}{*{20}{c}}{\bf{1}}&{\bf{3}}&{\bf{2}}&{ - {\bf{4}}}\\{\bf{0}}&{\bf{1}}&{\bf{2}}&{ - {\bf{5}}}\\{\bf{2}}&{\bf{7}}&{\bf{6}}&{ - {\bf{3}}}\\{ - {\bf{3}}}&{ - {\bf{10}}}&{ - {\bf{7}}}&{\bf{2}}\end{aligned}} \right|\)

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

Question: 6. Use Cramer’s rule to compute the solution of the following system.

\(\begin{array}{c}{x_{\bf{1}}} + {\bf{3}}{x_{\bf{2}}} + \,{x_{\bf{3}}} = {\bf{4}}\\ - {x_{\bf{1}}} + \,\,\,\,\,\,\,\,\,\,{\bf{2}}{x_{\bf{3}}} = {\bf{2}}\\{\bf{3}}{x_{\bf{1}}} + \,{x_{\bf{2}}}\,\,\,\,\,\,\,\,\,\,\,\,\, = {\bf{2}}\end{array}\)

Compute the determinant in Exercise 7 using a cofactor expansion across the first row.

7. \[\left| {\begin{array}{*{20}{c}}{\bf{4}}&{\bf{3}}&{\bf{0}}\\{\bf{6}}&{\bf{5}}&{\bf{2}}\\{\bf{9}}&{\bf{7}}&{\bf{3}}\end{array}} \right|\]

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

36. Let U be a square matrix such that \({U^T}U = I\). Show that\(det{\rm{ }}U = \pm 1\).
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

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