/*! 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} Q38Q Let \(A = \left( {\begin{aligned... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Q38Q (page 93)

Let \(A = \left( {\begin{aligned}{*{20}{c}}{\bf{1}}&{\bf{1}}&{\bf{1}}&{\bf{0}}\\{\bf{0}}&{\bf{1}}&{\bf{1}}&{\bf{1}}\end{aligned}} \right)\). Construct a \({\bf{4}} \times {\bf{2}}\) matrix \(D\) using only \({\bf{1}}\) and \({\bf{0}}\) as enteries, such that \(AD = {I_{\bf{2}}}\). It is possible that \(CA = {I_{\bf{4}}}\) for some \({\bf{4}} \times {\bf{2}}\) matrix \(C\)? Why or why not?

Short Answer

Expert verified

(D = \left( {\begin{aligned}{*{20}{c}}1&0\\0&0\\0&0\\0&1\end{aligned}} \right)\) and matrix \(C\) is not defined as the column vectors of \(A\) are linearly independent.

Step by step solution

01

Construct matrix \(D\) using the elements \({\bf{1}}\) and \({\bf{0}}\)

The first column of the matrix \(D\) is \(\left( {\begin{aligned}{*{20}{c}}1\\0\\0\\0\end{aligned}} \right)\).

When matrix \(A\) is multiplied by the column of \(D\), the first row of \(AD\) becomes \(\left( {\begin{aligned}{*{20}{c}}1&0\end{aligned}} \right)\).

02

Construct matrix \(D\) using the elements \({\bf{1}}\) and \({\bf{0}}\)

The second column of matrix \(D\) is \(\left( {\begin{aligned}{*{20}{c}}0\\0\\0\\1\end{aligned}} \right)\).

03

Find matrix \(C\)

If \(CA = {I_4}\), then \(CA{\bf{x}}\) is equal to \({\bf{x}}\) in \({\mathbb{R}^4}\). As the columns in \(A\) are linearly independent and \(A{\bf{x}} = 0\) for a non-zero vector \({\bf{x}}\), \(CA{\bf{x}}\) is not equal to \({\bf{x}}\).

So, \(D = \left( {\begin{aligned}{*{20}{c}}1&0\\0&0\\0&0\\0&1\end{aligned}} \right)\) and matrix \(C\) are not defined as the column vectors of \(A\) are linearly independent.

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

If Ais an \(n \times n\) matrix and the equation \(A{\bf{x}} = {\bf{b}}\) has more than one solution for some b, then the transformation \({\bf{x}}| \to A{\bf{x}}\) is not one-to-one. What else can you say about this transformation? Justify your answer.

A useful way to test new ideas in matrix algebra, or to make conjectures, is to make calculations with matrices selected at random. Checking a property for a few matrices does not prove that the property holds in general, but it makes the property more believable. Also, if the property is actually false, you may discover this when you make a few calculations.

37. Construct a random \({\bf{4}} \times {\bf{4}}\) matrix Aand test whether \(\left( {A + I} \right)\left( {A - I} \right) = {A^2} - I\). The best way to do this is to compute \(\left( {A + I} \right)\left( {A - I} \right) - \left( {{A^2} - I} \right)\) and verify that this difference is the zero matrix. Do this for three random matrices. Then test \(\left( {A + B} \right)\left( {A - B} \right) = {A^2} - {B^{\bf{2}}}\) the same way for three pairs of random \({\bf{4}} \times {\bf{4}}\) matrices. Report your conclusions.

If \(A = \left( {\begin{aligned}{*{20}{c}}1&{ - 2}\\{ - 2}&5\end{aligned}} \right)\) and \(AB = \left( {\begin{aligned}{*{20}{c}}{ - 1}&2&{ - 1}\\6&{ - 9}&3\end{aligned}} \right)\), determine the first and second column of B.

Prove Theorem 2(d). (Hint: The \(\left( {i,j} \right)\)- entry in \(\left( {rA} \right)B\) is \(\left( {r{a_{i1}}} \right){b_{1j}} + ... + \left( {r{a_{in}}} \right){b_{nj}}\).)

How many rows does \(B\) have if \(BC\) is a \({\bf{3}} \times {\bf{4}}\) matrix?
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Decision 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.