/*! 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 10 Compute the indicated products. ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 10

Compute the indicated products. $$ \left[\begin{array}{rrr} 3 & 2 & -1 \\ 4 & -1 & 0 \\ -5 & 2 & 1 \end{array}\right]\left[\begin{array}{r} 3 \\ -2 \\ 0 \end{array}\right] $$

Short Answer

Expert verified
The product of the given matrices is: $$ \left[\begin{array}{rrr} 3 & 2 & -1 \\ 4 & -1 & 0 \\ -5 & 2 & 1 \end{array}\right]\left[\begin{array}{r} 3 \\ -2 \\ 0 \end{array}\right] = \left[\begin{array}{r} 5 \\ 14 \\ -19 \end{array}\right] $$

Step by step solution

01

Set up the resulting matrix

We will first set up our resulting 3x1 matrix that will contain the final computed product. The dimensions of the resulting matrix will be 3x1. $$ \left[\begin{array}{r} 0 \\ 0 \\ 0 \end{array}\right] $$
02

Multiply rows by columns

Now we will multiply the corresponding elements of the rows in the first matrix with the columns in the second matrix, and then sum the results. The product will be filled in the resulting matrix. Row 1 of first matrix and the only column in the second matrix (3x1): $$ (3) * (3) + (2) * (-2) + (-1) * (0) = 9 - 4 + 0 = 5 $$ Row 2 of first matrix and the only column in the second matrix (3x1): $$ (4) * (3) + (-1) * (-2) + (0) * (0) = 12 + 2 + 0 = 14 $$ Row 3 of first matrix and the only column in the second matrix (3x1): $$ (-5) * (3) + (2) * (-2) + (1) * (0) = -15 - 4 + 0 = -19 $$
03

Fill the resulting matrix

Now that we have the products, we will fill the resulting matrix with the corresponding values. $$ \left[\begin{array}{r} 5 \\ 14 \\ -19 \end{array}\right] $$ The product of the given matrices is: $$ \left[\begin{array}{rrr} 3 & 2 & -1 \\ 4 & -1 & 0 \\ -5 & 2 & 1 \end{array}\right]\left[\begin{array}{r} 3 \\ -2 \\ 0 \end{array}\right] = \left[\begin{array}{r} 5 \\ 14 \\ -19 \end{array}\right] $$

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.

Linear Algebra
Linear algebra is a branch of mathematics focused on vector spaces and the linear mappings between these spaces. It includes the study of lines, planes, and subspaces, but is also concerned with properties common to all vector spaces.

When it comes to matrix multiplication in linear algebra, the operation can be understood as a way to transform one set of coordinates to another. This is crucial in various applications like 3D graphics, where matrices are used to rotate, scale, and translate points in space.

In the provided exercise, matrix multiplication transforms a 3-dimensional vector represented by a 3x1 matrix by a 3x3 matrix, which could, in a real-world scenario, represent rotation or scaling in three-dimensional space.
Matrix Operations
Matrix operations include a variety of computations, among which matrix multiplication is one of the most significant. Understanding how to multiply matrices is essential for solving many problems in mathematics and applied sciences.

Matrix multiplication, unlike multiplication of simple numbers, is not commutative. This means that the order in which you multiply matrices matters. Additionally, not all matrices can be multiplied together; the number of columns in the first matrix must match the number of rows in the second matrix.

The exercise demonstrates the process of multiplying a 3x3 matrix by a 3x1 matrix. The result is a 3x1 matrix, and the process involves the dot product of each row of the first matrix with the solitary column of the second matrix.
Mathematical Matrices
Mathematical matrices are rectangular arrays of numbers, symbols, or expressions arranged in rows and columns. The individual items in a matrix are called elements. The size of a matrix is defined by the number of rows and columns it possesses, denoted as 'm x n' where 'm' is the number of rows and 'n' is the number of columns.

Matrices can represent and solve systems of linear equations, serving as a compact way to deal with multiple equations at once. They are also the backbone of linear transformations in mathematics.

The exercise provided involves two matrices: a 3x3 matrix that could represent a system of linear equations or a transformation, and a 3x1 matrix that could represent a single point in space or a set of values. The multiplication of these matrices yields another 3x1 matrix, which could offer a new set of values or a transformed point in space.

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

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(A\) is a square matrix with inverse \(A^{-1}\) and \(c\) is a nonzero real number, then $$ (c A)^{-1}=\left(\frac{1}{c}\right) A^{-1} $$

A dietitian plans a meal around three foods. The number of units of vitamin A, vitamin \(\mathrm{C}\), and calcium in each ounce of these foods is represented by the matrix \(M\), where $$ \begin{array}{l} \text { Food I } & \text { Food II } & \text { Food III } \\ \text { Vitamin A } & {\left[\begin{array}{rrr} 400 & 1200 & 800 \\ M= & \text { Vitamin C } \\ \text { Calcium } \end{array}\right.} & \begin{array}{rr} 110 \\ 90 \end{array} & \begin{array}{r} 570 \\ 30 \end{array} & \left.\begin{array}{r} 340 \\ 60 \end{array}\right] \end{array} $$ The matrices \(A\) and \(B\) represent the amount of each food (in ounces) consumed by a girl at two different meals, where \(\begin{array}{lll}\text { Food I } & \text { Food II } & \text { Food III }\end{array}\) $$ A=\left[\begin{array}{lll} 7 & 1 & 6 \end{array}\right] $$ \(\begin{array}{lll}\text { Food I } & \text { Food II } & \text { Food III }\end{array}\) $$ B=\left[9 \quad \left[\begin{array}{ll} 9 & 3 \end{array}\right.\right. $$ $$ 2] $$ Calculate the following matrices and explain the meaning of the entries in each matrix. a. \(M A^{T}\) b. \(M B^{T}\) c. \(M(A+B)^{T}\)

Find the transpose of each matrix. $$ \left[\begin{array}{rrrr} 4 & 2 & 0 & -1 \\ 3 & 4 & -1 & 5 \end{array}\right] $$

A private investment club has a certain amount of money earmarked for investment in stocks. To arrive at an acceptable overall level of risk, the stocks that management is considering have been classified into three categories: high risk, medium risk, and low risk. Management estimates that high-risk stocks will have a rate of return of \(15 \% /\) year; medium-risk stocks, \(10 \% /\) year; and low-risk stocks, \(6 \%\) /year. The members have decided that the investment in low-risk stocks should be equal to the sum of the investments in the stocks of the other two categories. Determine how much the club should invest in each type of stock in each of the following scenarios. (In all cases, assume that the entire sum available for investment is invested.) a. The club has \(\$ 200,000\) to invest, and the investment goal is to have a return of \(\$ 20,000 /\) year on the total investment. b. The club has \(\$ 220,000\) to invest, and the investment goal is to have a return of \(\$ 22,000 /\) year on the total investment. c. The club has \(\$ 240,000\) to invest, and the investment goal is to have a return of \(\$ 22,000 /\) year on the total investment.

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(A\) and \(B\) are matrices of the same size and \(c\) is a scalar, then \(c(A+B)=c A+c B\).
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation

Probability and Statistics

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.