/*! 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 5 Perform the indicated multiplica... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 16: Problem 5

Perform the indicated multiplications. $$\left[\begin{array}{rrr}2 & -3 & 1 \\\0 & 7 & -3\end{array}\right]\left[\begin{array}{r}90 \\ -25 \\\50\end{array}\right]$$

Short Answer

Expert verified
The product of the matrices is \( \begin{bmatrix} 305 \\ -325 \end{bmatrix} \).

Step by step solution

01

Determine Matrix Dimensions

First, ensure that the matrices can be multiplied together by checking their dimensions. The first matrix is 2x3, and the second matrix is 3x1. Since the number of columns in the first matrix matches the number of rows in the second matrix, multiplication is possible.
02

Set Up the Multiplication

Set up the multiplication process by writing the matrix product in the form of a new matrix. The resulting matrix will have dimensions determined by the rows of the first matrix and the columns of the second matrix, which is 2x1 for this case.
03

Multiply Rows by Columns

For each entry in the resulting matrix, multiply the elements of the row from the first matrix with the corresponding elements of the column from the second matrix, and then find the sum of these products for each entry.
04

Calculate the First Element

Calculate the element in the first row and the first column of the resulting matrix: \((2 imes 90) + (-3 imes -25) + (1 imes 50) = 180 + 75 + 50 = 305\).
05

Calculate the Second Element

Calculate the element in the second row and the first column of the resulting matrix:\((0 imes 90) + (7 imes -25) + (-3 imes 50) = 0 - 175 - 150 = -325\).
06

Construct the Resulting Matrix

Combine the calculated elements into the resulting matrix. The product of the matrices will be:\[\begin{bmatrix} 305 \ -325 \end{bmatrix}\].

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 dimensions
Matrix dimensions are crucial when considering matrix multiplication. Each matrix has dimensions expressed as rows by columns. These dimensions determine whether two matrices can be multiplied. In our exercise, the first matrix is 2x3, meaning it has 2 rows and 3 columns. The second matrix is 3x1, with 3 rows and 1 column. For multiplication to occur, the number of columns in the first matrix must equal the number of rows in the second matrix. This condition is met here (3 columns = 3 rows). Therefore, multiplication is possible.
matrix product
The matrix product is the result of multiplying two matrices. The dimensions of the resulting matrix depend on the dimensions of the original matrices involved in the multiplication process. Specifically, the resulting matrix will have the number of rows of the first matrix and the number of columns of the second matrix. In our case, we are multiplying a 2x3 matrix with a 3x1 matrix. Thus, the resulting matrix will be a 2x1 matrix, having 2 rows and 1 column.
multiplication process
The multiplication process involves pairing each element in a row of the first matrix with each element in a column of the second matrix, multiplying these pairs, and summing the products. To start, take the first row of the first matrix:
  • Multiply it with the first element of the column from the second matrix and sum these products.
  • The calculation for the first element would be: \((2 \times 90) + (-3 \times -25) + (1 \times 50)\).
Repeat with the second row of the first matrix:
  • Multiply with the respective column elements of the second matrix.
  • The calculation for the second element is: \((0 \times 90) + (7 \times -25) + (-3 \times 50)\).
This detailed calculation forms each entry of the resulting matrix.
resulting matrix
The resulting matrix is the product of the multiplication process. After computing each sum of products, these values are placed as entries in the new matrix. In our example:
  • The first computed value, 305, becomes the first element in the matrix's first row and column.
  • The second computed value, -325, becomes the first (and only) element in the matrix's second row and first column.
As a result, the resulting matrix is:\[\begin{bmatrix} 305 \ -325 \end{bmatrix}\]This matrix accurately represents the combined influence of the original matrices through multiplication.

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

Solve the given problems by using determinants. An alloy is to be made from four other alloys containing copper (Cu), nickel (Ni), zinc (Zn), and Iron (Fe). The first is 80\% Cu and 20\% Ni. The second is 60\% Cu, 20\% Ni, and 20\% Zn. The third is \(30 \%\) Cu, \(60 \%\) Ni, and \(10 \%\) Fe. The fourth is \(20 \%\) Ni, \(40 \% \mathrm{Zn},\) and \(40 \% \mathrm{Fe} .\) How much of each is needed so that the final alloy has \(56 \mathrm{g} \mathrm{Cu}, 28 \mathrm{g} \mathrm{Ni}, 10 \mathrm{g} \mathrm{Zn},\) and \(6 \mathrm{g} \mathrm{Fe} ?\)

Solve the indicated systems of equations using the inverse of the coefficient matrix. It is necessary to set up the appropriate equations. A research chemist wants to make \(10.0 \mathrm{L}\) of gasoline containing \(2.0 \%\) of a new experimental additive. Gasoline without additive and two mixtures of gasoline with additive, one with \(5.0 \%\) and the other with \(6.0 \%,\) are to be used. If four times as much gasoline without additive as the \(5.0 \%\) mixture is to be used, how much of each is needed?

Perform the indicated matrix multiplications. Using Kirchhoff's laws on the circuit shown in Fig. 16.7 , the following matrix equation is found. By matrix multiplication, find the resulting system of equations. $$\left[\begin{array}{ccc}R_{1}+R_{2} & -R_{2} & 0 \\\\-R_{2} & R_{2}+R_{3}+R_{4} & -R_{4} \\ 0 & -R_{4} & R_{4}+R_{5}\end{array}\right]\left[\begin{array}{c}I_{1} \\\I_{2} \\\ I_{3}\end{array}\right]=\left[\begin{array}{c}V_{1} \\\0 \\\\-V_{2}\end{array}\right]$$

Perform the indicated matrix multiplications on a calculator, using the following matrices. For matrix \(A, A^{2}=A \times A.\) $$A=\left[\begin{array}{rrr}2 & -3 & -5 \\\\-1 & 4 & 5 \\\1 & -3 & -4\end{array}\right] B=\left[\begin{array}{rrr}1 & -2 & -6 \\\\-3 & 2 & 9 \\\2 & 0 & -3\end{array}\right] C=\left[\begin{array}{rrr}1 & -3 & -4 \\\\-1 & 3 & 4 \\\1 & -3 & -4\end{array}\right]$$ Show that \(C^{2}=O\)

Evaluate the given determinants by expansion by minors. $$\left|\begin{array}{rrrr} 6 & -3 & -6 & 3 \\ -2 & 1 & 2 & -1 \\ 18 & 7 & -1 & 5 \\ 0 & -1 & 10 & 10 \end{array}\right|$$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

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.