/*! 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 18 Let \(A=\left[\begin{array}{ll}4... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 18

Let \(A=\left[\begin{array}{ll}4 & -1 \\ 7 & -9\end{array}\right], B=\left[\begin{array}{rr}3 & 9 \\ 2 & -2\end{array}\right], C=\left[\begin{array}{rr}8 & 3 \\ 0 & -3\end{array}\right]\) \(D=\left[\begin{array}{rrr}5 & -6 & 1 \\ 10 & 3 & -1\end{array}\right], E=\left[\begin{array}{rr}-7 & 4 \\ -3 & 2 \\ 2 & -1\end{array}\right]\) \(F=\left[\begin{array}{rrr}-4 & 2 & 3 \\ 0 & -1 & 2 \\ -7 & -2 & 5\end{array}\right]\), and \(v=\left[\begin{array}{r}2 \\\ -3\end{array}\right]\). Compute (DE)A and \(\mathrm{D}\) (EA) to verify the associative property for these matrices.

Short Answer

Expert verified
Verify the associative property is true by showing (DE)A = D(EA).

Step by step solution

01

- Compute DE

Calculate the product of matrices D and E using matrix multiplication rules. Multiply each element of the rows of D with the corresponding elements of the columns of E and then sum these products.
02

- Compute (DE)A

First, write down the result from DE. Then, multiply the resulting matrix by the matrix A using the same rules for matrix multiplication.
03

- Compute EA

Compute the product of matrices E and A using the matrix multiplication rules.
04

- Compute D(EA)

First, write down the result from EA. Then, multiply the resulting matrix by the matrix D.
05

- Verify the Associative Property

Compare the results from steps 2 and 4. If both results are the same, then the associative property is verified for these matrices.

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 Multiplication
Matrix multiplication is a fundamental operation in linear algebra. It involves multiplying two matrices together to form a new matrix.
The product is calculated by taking the dot product of rows from the first matrix with the columns of the second matrix.
For example, to find the element in the first row and first column of the result matrix, you multiply each element of the first row of the first matrix by the corresponding element of the first column of the second matrix and sum them up.
\[ \text{If } A = \begin{bmatrix} a & b \ c & d \ \text{and } B = \begin{bmatrix} e & f \ g & h \ \text{then } AB = \begin{bmatrix} (ae+bg) & (af+bh) \ (ce+dg) & (cf+dh) \ \]
This technique is repeated for every corresponding row and column to fill the resulting matrix.
Matrix Associative Property
The associative property of matrix multiplication is an essential characteristic of matrices.
It states that when multiplying three or more matrices, the way in which the matrices are grouped does not change the product.
In other words, for any three matrices A, B, and C, the property is given by \[ (AB)C = A(BC) \ \]
This means that you can first multiply A and B, then multiply the resulting matrix by C, or you can first multiply B and C, then multiply the resulting matrix by A, and both will yield the same final matrix.
For example, if you have matrices D, E, and A, you can compute \[ (DE)A \]
or \[ D(EA) \]
and both will provide the same result. This property is very useful in simplifying complex matrix multiplication processes.
Linear Algebra
Linear algebra is the branch of mathematics concerning linear equations, linear functions, and their representations through matrices and vector spaces.
It encompasses a wide range of concepts, including vectors, vector spaces, linear transformations, and systems of linear equations.
Some key components of linear algebra include:
  • Scalars: Single numbers used for multiplication.
  • Vectors: Ordered lists of numbers.
  • Matrices: Rectangular arrays of numbers.
  • Linear transformations: Functions that map vectors to vectors in a linear manner.
One of the most powerful tools in linear algebra is the matrix, which can be used to represent and solve systems of linear equations.
Understanding the properties and operations of matrices, like the associative property of matrix multiplication, is crucial for many applications in science, engineering, and computer science.

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 \(A=\left[\begin{array}{ll}4 & -1 \\ 7 & -9\end{array}\right], B=\left[\begin{array}{rr}3 & 9 \\ 2 & -2\end{array}\right], C=\left[\begin{array}{rr}8 & 3 \\ 0 & -3\end{array}\right]\) \(D=\left[\begin{array}{rrr}5 & -6 & 1 \\ 10 & 3 & -1\end{array}\right], E=\left[\begin{array}{rr}-7 & 4 \\ -3 & 2 \\ 2 & -1\end{array}\right]\) \(F=\left[\begin{array}{rrr}-4 & 2 & 3 \\ 0 & -1 & 2 \\ -7 & -2 & 5\end{array}\right]\), and \(v=\left[\begin{array}{r}2 \\\ -3\end{array}\right]\). Compute \(A-B\) and \(A+(-1)\) B. Compare your answers.

The matrix \(A\) has eigenvalues \(r_{1}\) and \(r_{2}\) with corresponding eigenvectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\), respectively. Compute \(\mathbf{A}^{n} \mathbf{w}\). $$ \begin{array}{l} r_{1}=3, r_{2}=0, v_{1}=\left[\begin{array}{r} 1 \\ -1 \end{array}\right], v_{2}=\left[\begin{array}{l} 1 \\ 1 \end{array}\right], n=4, \\ w=\left[\begin{array}{l} 2 \\ 4 \end{array}\right] \end{array} $$

Write the vector \(v\) as a linear combination of the vectors \(\mathbf{w}\) and \(\mathbf{u}\). $$ \mathbf{v}=\left[\begin{array}{l} 3 \\ 4 \end{array}\right], \mathbf{w}=\left[\begin{array}{l} 1 \\ 0 \end{array}\right], \mathbf{u}=\left[\begin{array}{l} 0 \\ 1 \end{array}\right] $$

Dinophilus gyrociliatus is a small species that lives in the fouling community of harbor environments. On average, a female has approximately 30 eggs during her first 6 wk of life. If she survives her first \(6 \mathrm{wk}\), she has on average 15 eggs her second 6 wk of life. Furthermore, approximately \(80 \%\) of the females survive their first \(6 \mathrm{wk}\) and none survive beyond the second \(6 \mathrm{wk} .{ }^{+}\) Assume half the eggs are female and for simplicity, assume that all the eggs are hatched at once at the beginning of each 6-wk period. Ignore the male population and make the two groups females under 6 wk old and females over 6 wk old. a) Draw and label the Leslie diagram. b) Find the Leslie matrix. c) Twenty hatchlings are introduced into an area. Estimate the population of the two groups after 6 wk. d) Estimate the population of the two groups after 12 wk.

Multiply the matrix and the vector to determine if the vector is an eigenvector. If so, what is the eigenvalue? $$ \left[\begin{array}{cc} -8.5 & -4.5 \\ 21 & 11 \end{array}\right],\left[\begin{array}{r} 2 \\ -7 \end{array}\right] $$
See all solutions

Recommended explanations on Biology Textbooks

Reproduction

Read Explanation

Cell Cycle

Read Explanation

Astrobiological Science

Read Explanation

Ecological Levels

Read Explanation

Responding to Change

Read Explanation

Biological Structures

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.