Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 2: Q16SE (page 93)

Let A be an \(n \times n\) singular matrix. Describe how to construct an \(n \times n\) nonzero matrix B such that \(AB = 0\).

Short Answer

Expert verified

It is shown that \(AB = 0\).

Step by step solution

01

Explain the construction of a \(n \times n\) non-zero matrix

There is a non-zero vector v in \({\mathbb{R}^n}\) such that \(A{\mathop{\rm v}\nolimits} = 0\) because A is not invertible. Put \(n\) copies of vector \({\mathop{\rm v}\nolimits} \) in a \(n \times n\) matrix B. It gives

\(\begin{array}{c}AB = A\left( {\begin{array}{*{20}{c}}{\mathop{\rm v}\nolimits} & \ldots &{\mathop{\rm v}\nolimits} \end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}{A{\mathop{\rm v}\nolimits} }& \ldots &{A{\mathop{\rm v}\nolimits} }\end{array}} \right)\\ = 0.\end{array}\)

Thus, it is shown that \(AB = 0\).

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.

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

2. Find the inverse of the matrix \(\left( {\begin{aligned}{*{20}{c}}{\bf{3}}&{\bf{2}}\\{\bf{7}}&{\bf{4}}\end{aligned}} \right)\).

Suppose the third column of Bis the sum of the first two columns. What can you say about the third column of AB? Why?

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.

36. Write the command(s) that will create a \(6 \times 4\) matrix with random entries. In what range of numbers do the entries lie? Tell how to create a \(3 \times 3\) matrix with random integer entries between \( - {\bf{9}}\) and 9. (Hint:If xis a random number such that 0 < x < 1, then \( - 9.5 < 19\left( {x - .5} \right) < 9.5\).

Generalize the idea of Exercise 21(a) [not 21(b)] by constructing a \(5 \times 5\) matrix \(M = \left[ {\begin{array}{*{20}{c}}A&0\\C&D\end{array}} \right]\) such that \({M^2} = I\). Make C a nonzero \(2 \times 3\) matrix. Show that your construction works.

In Exercises 33 and 34, Tis a linear transformation from \({\mathbb{R}^2}\) into \({\mathbb{R}^2}\). Show that T is invertible and find a formula for \({T^{ - 1}}\).

33. \(T\left( {{x_1},{x_2}} \right) = \left( { - 5{x_1} + 9{x_2},4{x_1} - 7{x_2}} \right)\)
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

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