/*! 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} Q5.2-26E Question: Let \(A = \left( {\beg... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Q5.2-26E (page 267)

Question: Let \(A = \left( {\begin{array}{*{20}{c}}a&b\\c&d\end{array}} \right)\). Use formula (1) for a determinant (given before Example 2) to show that \(\det A = ad - bc\). Consider two cases: \(a \ne 0\) and \(a = 0\).

Short Answer

Expert verified

It is proved that \({\rm{det}}A = ad - cb\) when \(a \ne 0\) and \(a = 0\).

Step by step solution

01

Consider \(a \ne 0\) and determine the determinant

Consider \(a \ne 0\), apply the row operations on the matrix \(A\).

\[\begin{array}A = \left( {\begin{array}{*{20}{c}}a&b\\c&d\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}a&b\\0&{d - c{a^{ - 1}}b}\end{array}} \right)\;\;\;\;\;\;\;\;\;\;{R_2} \to {R_2} - \frac{1}{a}{R_1}\\ = U\end{array}\]

Find the determinant of \(A\).

\(\begin{array}{\rm{det}}A = a\left( {d - c{a^{ - 1}}b} \right) - b\left( 0 \right)\\ = ad - cb\end{array}\)

Thus, it is proved that \({\rm{det}}A = ad - cb\) when \(a \ne 0\).

02

Consider \(a = 0\) and determine the determinant

Consider \(a = 0\), apply the row operations on the matrix \(A\).

\[\begin{array}A = \left( {\begin{array}{*{20}{c}}0&b\\c&d\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}c&d\\0&b\end{array}} \right)\;\;\;\;\;\;\;\;\;\;{R_2} \leftrightarrow {R_1}\\ = U\end{array}\]

Find the determinant of \(A\).

\[\begin{array}{\rm{det}}A = {\left( { - 1} \right)^1}\left( {cb} \right)\\ = 0 - bc\\ = ad - bc\end{array}\]

Thus, it is proved that \({\rm{det}}A = ad - cb\) when \(a = 0\).

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

Question: Diagonalize the matrices in Exercises \({\bf{7--20}}\), if possible. The eigenvalues for Exercises \({\bf{11--16}}\) are as follows:\(\left( {{\bf{11}}} \right)\lambda {\bf{ = 1,2,3}}\); \(\left( {{\bf{12}}} \right)\lambda {\bf{ = 2,8}}\); \(\left( {{\bf{13}}} \right)\lambda {\bf{ = 5,1}}\); \(\left( {{\bf{14}}} \right)\lambda {\bf{ = 5,4}}\); \(\left( {{\bf{15}}} \right)\lambda {\bf{ = 3,1}}\); \(\left( {{\bf{16}}} \right)\lambda {\bf{ = 2,1}}\). For exercise \({\bf{18}}\), one eigenvalue is \(\lambda {\bf{ = 5}}\) and one eigenvector is \(\left( {{\bf{ - 2,}}\;{\bf{1,}}\;{\bf{2}}} \right)\).

8. \(\left( {\begin{array}{*{20}{c}}{\bf{5}}&{\bf{1}}\\{\bf{0}}&{\bf{5}}\end{array}} \right)\)

Use Exercise 12 to find the eigenvalues of the matrices in Exercises 13 and 14.

13. \(A = \left( {\begin{array}{*{20}{c}}3&{ - 2}&8\\0&5&{ - 2}\\0&{ - 4}&3\end{array}} \right)\)

Suppose \(A\) is diagonalizable and \(p\left( t \right)\) is the characteristic polynomial of \(A\). Define \(p\left( A \right)\) as in Exercise 5, and show that \(p\left( A \right)\) is the zero matrix. This fact, which is also true for any square matrix, is called the Cayley-Hamilton theorem.

Question: In Exercises \({\bf{1}}\) and \({\bf{2}}\), let \(A = PD{P^{ - {\bf{1}}}}\) and compute \({A^{\bf{4}}}\).

1. \(P{\bf{ = }}\left( {\begin{array}{*{20}{c}}{\bf{5}}&{\bf{7}}\\{\bf{2}}&{\bf{3}}\end{array}} \right)\), \(D{\bf{ = }}\left( {\begin{array}{*{20}{c}}{\bf{2}}&{\bf{0}}\\{\bf{0}}&{\bf{1}}\end{array}} \right)\)

(M)The MATLAB command roots\(\left( p \right)\) computes the roots of the polynomial equation \(p\left( t \right) = {\bf{0}}\). Read a MATLAB manual, and then describe the basic idea behind the algorithm for the roots command.
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Logic and Functions

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.