Log In Start studying!

Select your language

Suggested languages for you:
Deutsch (DE)
Deutsch (UK)
Deutsch (US)

Americas

Europe

Answers without the blur. Sign up and see all textbooks for free! → Illustration

Q20E

Expert-verified
Linear Algebra and its Applications
Found in: Page 165
Linear Algebra and its Applications

Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

Answers without the blur.

Just sign up for free and you're in.

Register for Free I’ll do it later
Illustration

Short Answer

Question: In Exercise 20, find the area of the parallelogram whose vertices are listed.

20. \(\left( {0,0} \right),\left( { - {\bf{2}},4} \right),\left( {4, - 5} \right),\left( {2, - 1} \right)\)

The area of the parallelogram is 2 square units.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Determine the matrix

The column vectors in the parallelogram are \(\left( {\begin{array}{*{20}{c}}0\\0\end{array}} \right),\left( {\begin{array}{*{20}{c}}{ - 2}\\4\end{array}} \right),\left( {\begin{array}{*{20}{c}}4\\{ - 5}\end{array}} \right),\) and \(\left( {\begin{array}{*{20}{c}}2\\{ - 1}\end{array}} \right)\).

Note that the parallelogram has origin as a vertex and

\(\begin{array}{c}\left( {\begin{array}{*{20}{c}}{ - 2}\\4\end{array}} \right) + \left( {\begin{array}{*{20}{c}}4\\{ - 5}\end{array}} \right) = \left( {\begin{array}{*{20}{c}}{ - 2 + 4}\\{4 - 5}\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}2\\{ - 1}\end{array}} \right).\end{array}\)

Hence, this parallelogram is determined by the columns of \(A = \left( {\begin{array}{*{20}{c}}{ - 2}&4\\4&{ - 5}\end{array}} \right)\).

Step 2: Write the first statement of Theorem 9

The first statement of Theorem 9 states if A is a \(2 \times 2\) matrix, the area of the parallelogram determined by the columns of A is \(\left| {\det A} \right|\).

Step 3: Find the area

By the above statement,

\(\begin{array}{c}\left| {\det A} \right| = \left| {\det \left[ {\begin{array}{*{20}{c}}{ - 2}&4\\4&{ - 5}\end{array}} \right]} \right|\\ = \left| {10 - 8} \right|\\ = \left| 2 \right|\\\left| {\det A} \right| = 2\end{array}\)

Hence, the area of the parallelogram is 2 square units.

Most popular questions for Math Textbooks

Compute the determinant in Exercise 2 using a cofactor expansion across the first row. Also compute the determinant by a cofactor expansion down the second column.

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

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.

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

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

  1. \(\begin{array}{l}5{x_1} + 7{x_2} = 3\\2{x_1} + 4{x_2} = 1\end{array}\)

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

28. \(\left( {\begin{aligned}{*{20}{c}}k&0&0\\0&1&0\\0&0&1\end{aligned}} \right)\).

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

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

Want to see more solutions like these?

Sign up for free to discover our expert answers
Get Started - It’s free

Recommended explanations on Math Textbooks

Decision Maths

Read explanation

Calculus

Read explanation

Probability and Statistics

Read explanation

Statistics

Read explanation

Mechanics Maths

Read explanation

Geometry

Read explanation

Pure Maths

Read explanation

94% of StudySmarter users get better grades.

Sign up for free
94% of StudySmarter users get better grades.