/*! 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} Problem 83 If \(A B=-B A,\) then \(A\) and ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 83

If \(A B=-B A,\) then \(A\) and \(B\) are said to be anticommutative. Are \(A=\left[\begin{array}{rr}0 & -1 \\ 1 & 0\end{array}\right]\) and \(B=\left[\begin{array}{rr}1 & 0 \\ 0 & -1\end{array}\right]\) anticommutative?

Short Answer

Expert verified
Yes, matrices A and B are anticommutative because the product AB equals the negation of the product BA.

Step by step solution

01

Multiply A and B

The first operation is to multiply matrices \(A\) and \(B\). This will provide the matrix \(AB\). Use standard matrix multiplication techniques. Each element in the product matrix is the dot product of its corresponding row in the first matrix and column in the second matrix.\r\n \( A \cdot B = \left[ \begin{array}{cc} 0 & -1 \ 1 & 0 \end{array} \right] \cdot \left[ \begin{array}{cc} 1 & 0 \ 0 & -1 \end{array} \right] = \left[ \begin{array}{cc} 0 & 1 \ 1 & 0 \end{array} \right] = AB \)
02

Multiply B and A

Next, do the multiplication in the opposite order to get \(BA\). Note that in general, the result will be different due to the non-commutative property of matrix multiplication: \( BA = \left[ \begin{array}{cc} 1 & 0 \ 0 & -1 \end{array} \right] \cdot \left[ \begin{array}{cc} 0 & -1 \ 1 & 0 \end{array} \right] = \left[ \begin{array}{cc} 0 & -1 \ -1 & 0 \end{array} \right] = BA. \)
03

Check if AB equals -BA

Now we negate \(BA\) and compare with \(AB\) to see if \( A \) and \( B \) are anticommutative. Negating \( BA \) we have \( -BA = \left[ \begin{array}{cc} 0 & 1 \ 1 & 0 \end{array} \right] \). Comparing with the result of step 1 (i.e. \( AB \)), we see that they are equal, thus \( A \) and \( B \) are indeed anticommutative.

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

Describe how to subtract matrices.

Write each system in the form \(A X=B\). Then solve the system by entering \(A\) and \(B\) into your graphing utility and computing \(A^{-1} B\). $$\left\\{\begin{array}{l}3 x-2 y+z=-2 \\\4 x-5 y+3 z=-9 \\\2 x-y+5 z=-5\end{array}\right.$$

Use a graphing utility to find the multiplicative inverse of each matrix. Check that the displayed inverse is correct. $$\left[\begin{array}{rr}3 & -1 \\\\-2 & 1\end{array}\right]$$

Low-resolution digital photographs use \(262,144\) pixels in a \(512 \times 512\) grid. If you enlarge a low-resolution digital photograph enough, describe what will happen.

Exercises \(77-79\) will help you prepare for the material covered in the first section of the next chapter. Consider the equation \(\frac{x^{2}}{9}+\frac{y^{2}}{4}-1\) a. Set \(y-0\) and find the \(x\) -intercepts. b. Set \(x-0\) and find the \(y\) -intercepts.
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

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.