/*! 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 36 If possible, find \(A B\) and st... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 36

If possible, find \(A B\) and state the order of the result. $$A=\left[\begin{array}{rrr} 0 & -1 & 2 \\ 6 & 0 & 3 \\ 7 & -1 & 8 \end{array}\right], \quad B=\left[\begin{array}{rr} 2 & -1 \\ 4 & -5 \\ 1 & 6 \end{array}\right]$$

Short Answer

Expert verified
Yes, it is possible to find \(AB\) and its order is 3x2. The resultant matrix is \(\left[\begin{array}{rr} -4 & -8 \ 18 & 12 \ 36 & 43 \end{array}\right]\).

Step by step solution

01

Check if multiplication is possible

Check if the number of columns in the first matrix, \(A\), is equal to the number of rows in the second matrix, \(B\). In this case, both are 3. So, the matrices are conformable for multiplication.
02

Determine the order of the resultant matrix

The resultant matrix will have the same number of rows as the first matrix, \(A\), and the same number of columns as the second matrix, \(B\). Therefore, the resultant matrix will be of order 3x2.
03

Perform the multiplication

Each element in the resultant matrix is calculated as the sum of the product of the corresponding elements in the row of the first matrix and the column of the second matrix. For example, the element in the first row and first column of the resultant matrix is calculated as follows: (0*2) + (-1*4) + (2*1) = -4. Performing these calculations for all the rows and columns of the resultant matrix gives the following matrix: \[\left[\begin{array}{rr} -4 & -8 \ 18 & 12 \ 36 & 43 \end{array}\right]\]

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Matrix Order
When working with matrices, understanding the matrix order is crucial. The order of a matrix refers to its dimensions, which are typically described by the number of rows and columns it contains. For instance, a matrix with 2 rows and 3 columns is said to be of order \(2 \times 3\).

In matrix multiplication, knowing the order of the matrices involved helps to determine if the operation is feasible and what the resulting matrix will look like. Given matrices \(A\) and \(B\) in the original exercise, where \(A\) is of order \(3 \times 3\) and \(B\) is of order \(3 \times 2\), understanding their orders allows us to proceed with the next step of matrix multiplication efficiently.
Conformable Matrices
A key requirement for multiplying two matrices is that they must be conformable. This means that the number of columns in the first matrix must be equal to the number of rows in the second matrix. Ensuring that matrices are conformable is the first check any time you are tasked with multiplying matrices.

In our example, matrix \(A\) has 3 columns, and matrix \(B\) has 3 rows, making them conformable for multiplication. Knowing this, we proceed, confident that a meaningful resultant matrix can be derived from the multiplication process. Failing to use conformable matrices would mean the multiplication isn't possible and would result in a mathematical error.
Resultant Matrix
Once matrices are confirmed to be conformable, the next interesting concept is the resultant matrix from their multiplication. The resultant matrix takes the form of the number of rows from the first matrix and the number of columns from the second matrix.

From our exercise, multiplying \(A\) (\(3 \times 3\)) by \(B\) (\(3 \times 2\)) results in a matrix of order \(3 \times 2\). Each element in this resultant matrix is computed as the sum of the products of corresponding elements in a row of the first matrix and a column of the second matrix. For instance, the top-left element is obtained through a calculation such as \((0 \times 2) + (-1 \times 4) + (2 \times 1) = -4\), contributing to a complete understanding of matrix multiplication dynamics.

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

Let \(A\) and \(B\) be unequal diagonal matrices of the same order. (A diagonal matrix is a square matrix in which each entry not on the main diagonal is zero.) Determine the products \(A B\) for several pairs of such matrices. Make a conjecture about a quick rule for such products.

Consider the circuit shown in the figure. The currents \(I_{1}, I_{2},\) and \(I_{3}\), in amperes, are the solution of the system of linear equations. $$\left\\{\begin{aligned}2 I_{1}\quad\quad\quad &+4 I_{3}=E_{1} \\\I_{2}+4 I_{3} &=E_{2} \\\I_{1}+I_{2}-I_{3} &=0\end{aligned}\right.$$ where \(E_{1}\) and \(E_{2}\) are voltages. Use the inverse of the coefficient matrix of this system to find the unknown currents for the given voltages. $$\begin{aligned}&E_{1}=28 \text { volts }\\\&E_{2}=21 \text { volts }\end{aligned}$$

At a local dairy mart, the numbers of gallons of skim milk, \(2 \%\) milk, and whole milk sold over the weekend are represented by \(A\). $$A=\left[\begin{array}{lll} 40 & 64 & 52 \\ 60 & 82 & 76 \\ 76 & 96 & 84 \end{array}\right]$$ The selling prices (in dollars per gallon) and the profits (in dollars per gallon) for the three types of milk sold by the dairy mart are represented by \(B\). $$B=\left[\begin{array}{ll} \$ 3.45 & \$ 1.20 \\ \$ 3.65 & \$ 1.30 \\ \$ 3.85 & \$ 1.45 \end{array}\right]$$ (a) Compute \(A B\) and interpret the result. (b) Find the dairy mart's total profit from milk sales for the weekend.

Determine whether the statement is true or false. Justify your answer. Two matrices can be added only when they have the same order.

Let \(i=\sqrt{-1}\) and let $$A=\left[\begin{array}{ll} i & 0 \\ 0 & i \end{array}\right] \quad \text { and } \quad B=\left[\begin{array}{cc} 0 & -i \\ i & 0 \end{array}\right].$$ (a) Find \(A^{2}, A^{3},\) and \(A^{4} .\) Identify any similarities with \(i^{2}\) \(i^{3},\) and \(i^{4}\). (b) Find and identify \(B^{2}\).
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

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.