/*! 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 3 Man bestimme den Rang der Matriz... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 3

Man bestimme den Rang der Matrizen (vgl. auch den Absch nitt 1.6.3) A. \(\left(\begin{array}{rrrrr}0 & 3 & 6 & 9 & -3 \\ 0 & 0 & 2 & -4 & 8 \\ 1 & 2 & -3 & 4 & 5 \\ 1 & 5 & 5 & 9 & 10\end{array}\right)\). B. \(\left(\begin{array}{rrrr}1 & -3 & 2 & 0 \\ 4 & -11 & 10 & -1 \\ -2 & 8 & -5 & 3\end{array}\right)\)

Short Answer

Expert verified
The rank of matrix A is 3, and the rank of matrix B is 2.

Step by step solution

01

Write matrix A

Consider matrix A given by: \[A = \begin{pmatrix} 0 & 3 & 6 & 9 & -3 \ 0 & 0 & 2 & -4 & 8 \ 1 & 2 & -3 & 4 & 5 \ 1 & 5 & 5 & 9 & 10 \end{pmatrix}.\]
02

Perform Row Operations on Matrix A

Perform row operations on matrix A to bring it to row echelon form. Start by swapping rows 1 and 3 to get a leading 1 in the first column.Next, aim to zero out elements below the first pivot in the first column by replacing the fourth row with \( R_4 - R_3 \).Continue by making the second and third rows zeros in the second column using suitable operations.
03

Determine the Row Echelon Form of Matrix A

By continuing with the row operations, you should form:\[\begin{pmatrix} 1 & 2 & -3 & 4 & 5 \ 0 & 3 & 6 & 9 & -3 \ 0 & 0 & 2 & -4 & 8 \ 0 & 0 & 0 & 0 & 0 \end{pmatrix}.\]
04

Identify the Non-zero Rows in Matrix A

Count the number of non-zero rows in the row-echelon form. Here, there are three non-zero rows.
05

Determine Rank of Matrix A

The rank of matrix A is equal to the number of non-zero rows, which is 3.
06

Write matrix B

Consider matrix B given by: \[B = \begin{pmatrix} 1 & -3 & 2 & 0 \ 4 & -11 & 10 & -1 \ -2 & 8 & -5 & 3 \end{pmatrix}.\]
07

Perform Row Operations on Matrix B

First, prepare the matrix by turning each element below the first pivot in the first column to zero. Replace the second row with \( R_2 - 4R_1 \) and the third row with \( R_3 + 2R_1 \).Continue this process to bring B to row echelon form.
08

Determine the Row Echelon Form of Matrix B

The row echelon form after operations is:\[\begin{pmatrix} 1 & -3 & 2 & 0 \ 0 & 1 & 2 & -1 \ 0 & 0 & 0 & 0 \end{pmatrix}.\]
09

Identify the Non-zero Rows in Matrix B

Count the non-zero rows in the row-echelon form. In matrix B, there are two non-zero rows.
10

Determine Rank of Matrix B

The rank of matrix B is the number of non-zero rows, which is 2.

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.

Row Echelon Form
The row echelon form of a matrix is an essential concept in linear algebra, especially when determining the rank of a matrix. This form serves as a simplified representation of a matrix achieved through row operations. In the row echelon form,
  • All non-zero rows are above any rows containing only zeros.
  • The leading entry of each non-zero row (known as a pivot) is strictly to the right of the leading entry of the row above it.
  • Below a pivot, all elements are zeros.
Converting a matrix to its row echelon form involves using elementary row operations, which can make matrix calculations much simpler. For example, when transforming matrix A in the exercise, a sequence of row swaps and linear combinations of rows helped achieve this crucial form.
Row Operations
Row operations are fundamental tools in matrix manipulation for various applications in linear algebra. They include:
  • Swapping: Exchanging two rows within a matrix helps position pivots correctly.
  • Scaling: Multiplying a row by a non-zero scalar adjusts the row for easier pivot management.
  • Addition or Subtraction: Adding or subtracting a multiple of one row from another helps eliminate specific elements, aiding in zeroing out elements below a pivot.
Using these operations is crucial when converting a matrix into row echelon form, as seen in the exercise. By systematically applying these changes, you can manipulate a matrix into a more straightforward and useful form for problem-solving.
Linear Algebra
Linear algebra is a branch of mathematics that deals with vectors, vector spaces, and linear transformations. One of its core focuses involves studying matrices and operations applicable to them, such as finding ranks. The rank of a matrix indicates the number of independent rows or columns, which reflects the maximum number of linearly independent vectors in the row or column space.
In the exercise, the rank of a matrix is derived by reducing matrices to their row echelon form. Identifying the number of non-zero rows provides the rank, revealing essential properties about solutions to systems of linear equations and the possible transformations by the matrix.
Matrix Transformation
Matrix transformation in linear algebra refers to the action of a matrix upon vectors in a space, thus altering their position or orientation. In essence, matrices can transform sets of data when applied as linear mappings. Key points to understand about matrix transformations:
  • Linear Transformation: A function that maps vectors to new locations in a linear manner, maintaining vector addition and scalar multiplication relationships.
  • Effect on Vector Space: Transformations can scale, rotate, reflect, or shear vectors in space.
  • Rank Relation: The rank of a matrix determines how completely it can transform the entire space; lower rank indicates the loss of dimensions in the resulting image of transformation.
The exercise illustrates transformations through row operations, rearranging the data to reveal possible ranks which highlight the transformations a matrix can impose on a vector 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

Man invertiere die beiden folgenden Matrizen und mache die Probe, d. h. zeige, dass das Produkt der Ausgangsmatrix mit der berech neten Inversen die Ein heits matrix ergibt (vgl. dazu auch Abschnitt 1.6.2). A. \(\left(\begin{array}{lll}1 & 0 & 2 \\ 2 & 1 & 3 \\ 1 & 1 & 2\end{array}\right)\) B. \(\left(\begin{array}{rrrr}0 & 1 & 1 & -1 \\ 1 & 2 & -1 & 1 \\ 1 & 3 & -1 & 0 \\ -1 & -2 & 1 & -2\end{array}\right) .\)

Aufgabe \(1.4\) Man berechne die beiden folgenden Determinanten (vgl. dazu auch Abschnitt 1.6.1): A. \(\Delta=\left|\begin{array}{ccc}1 & 0 & -1 \\ 3 & 1 & -3 \\ 1 & 2 & -2\end{array}\right|\) B. \(\Delta=\left|\begin{array}{rrrr}1 & 2 & -4 & 4 \\ 3 & 6 & 9 & 0 \\ -2 & 8 & 1 & 9 \\ 4 & 2 & -2 & 0\end{array}\right|\).

Man schreibe f眉r \(N=3\) die folgenden Ausd r?cke aus: A. \(v_{i}=\varepsilon_{i j k} \omega_{j} r_{k}\) Es seien \(v_{i}, \omega_{i}\) und \(r_{i}\) die kartesischen Koordinaten d reier Vektoren \(\underline{v}, \underline{\omega}\) und \(\underline{r}\). Wie wird der durch die obige Gleichu ng gegebene vektoralgebraische Zusammenhang zwischen \(\underline{v}, \underline{\omega}\) und \(\underline{r}\) 眉blicherweise gesch rieben? B. \(\Omega_{j}=\varepsilon_{i j k} \frac{\partial v_{i}}{\partial x_{k}}\). Es seien \(v_{i}\) und \(\Omega_{i}\) die kartesischen Koordinaten zweier Feldvektoren \(\underline{v}\) und \(\underline{\Omega}\), d. h. die \(v_{i}\) und die \(\Omega_{i}\) seien Funktionen der Ortskoord inaten \(x_{i}=\) \(\left(x_{1}, x_{2}, x_{3}\right)=(x, y, z)\). Wie wird der durch die obige Gleichung gegebene vektoranalyt ische Zusammenh ang zwischen \(\underline{\Omega}\) und \(\underline{v}\) 眉blicherweise gesch rieben?

Man bestimme den Rang der Matrizen (vgl. auch den Absch nitt 1.6.3) A. \(\left(\begin{array}{rrrrr}0 & 3 & 6 & 9 & -3 \\ 0 & 0 & 2 & -4 & 8 \\ 1 & 2 & -3 & 4 & 5 \\ 1 & 5 & 5 & 9 & 10\end{array}\right)\). B. \(\left(\begin{array}{rrrr}1 & -3 & 2 & 0 \\ 4 & -11 & 10 & -1 \\ -2 & 8 & -5 & 3\end{array}\right)\)

A. Sind die folgenden Gleichu ngen nach der Summationskon vention zul盲ssig? 1\. \(A_{i} B_{j j}=C_{i} D_{k k}\), 2\. \(A_{i} B_{i}=C_{i} D_{j}\) 3\. \(A_{i} B_{i}=C_{i} D_{\underline{i}}\), 4\. \(A_{\underline{\underline{i}}} B_{i}=C_{i}\) 5\. \(A_{m} B_{n}=C_{m n}\) 6\. \(A_{i} B_{k}=C_{i} D_{j}\), 7\. \(\quad A_{i} B_{k}=A_{k} B_{i}\) 8\. \(\mu_{m n} A_{m}=c_{n} F_{i i}\) 9\. \(A=\alpha_{\underline{m}} C_{m m}\), 10\. \(A_{i i}=B_{i j} C_{j j i}\). B. Man schreibe die zul盲ssigen Gleichungen f眉r \(N=2\) aus.
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

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.