/*! 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 7 Express the invertible matrix ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 7

Express the invertible matrix $$ \left(\begin{array}{lll} 1 & 2 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 2 \end{array}\right) $$ as a product of elementary matrices.

Short Answer

Expert verified
The given matrix can be expressed as the product of elementary matrices as follows: \( E_1 * E_2 * E_3 = \left( \begin{array}{lll} 1 & 2 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 2 \end{array} \right)\), where \(E_1 = \left( \begin{array}{lll} 2 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right)\), \( E_2 = \left( \begin{array}{lll} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right)\), and \( E_3 = \left( \begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & -1 & 1 \end{array} \right)\).

Step by step solution

01

Start with Identity Matrix

We start with an identity matrix: \[ I = \left( \begin{array}{lll} 1 & 0 & 0\\\ 0 & 1 & 0\\\ 0 & 0 & 1 \end{array} \right) \]
02

Define Row Operations

Define the row operations required to transform the identity matrix into the given matrix. We can see that: 1. Row1 = Row2 + 2*Row1 2. Row2 = Row1 - Row3 3. Row3 remains same. Write down the elementary matrices corresponding to these operations: 1. \( E_1 = \left( \begin{array}{lll} 2 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right)\) 2. \( E_2 = \left( \begin{array}{lll} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right)\) 3. \( E_3 = \left( \begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & -1 & 1 \end{array} \right)\)
03

Discover the Product of Elementary Matrices

Now, the given matrix can be expressed as the product of these elementary matrices. This results in: \( E_1 * E_2 * E_3 = \left( \begin{array}{lll} 1 & 2 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 2 \end{array} \right)\) That completes the task of expressing the given invertible matrix as a product of elementary 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 Inversion
Matrix inversion is a method used to find a matrix that, when multiplied by the original matrix, produces the identity matrix. This concept is closely related to finding the inverse of a matrix. Not every matrix has an inverse; only invertible matrices do. A matrix is invertible if it is square (same number of rows and columns) and its determinant is not zero.

The inverse of a matrix \( A \) is denoted by \( A^{-1} \), and it satisfies the equation \( A \cdot A^{-1} = I \), where \( I \) is the identity matrix. The process of finding the inverse involves applying row operations to transform \( A \) into the identity matrix while performing the identical operations on an identity matrix of similar size to ascertain \( A^{-1} \). This way, row operations do not alter the dimensions of the matrix.
  • Inversion is only possible if the matrix is non-singular, meaning its determinant is not zero.
  • The inverse is used to solve matrix equations, akin to finding reciprocals in algebra.
Row Operations
Row operations are vital tools in linear algebra that can modify matrices to solve systems of equations or find inverses. There are three types of row operations that can be performed on a matrix:
  • Row Switching: swapping two rows.
  • Row Multiplication: multiplying all entries of a row by a non-zero scalar.
  • Row Addition: adding a multiple of one row to another row.

These operations come into play especially when simplifying a matrix or transforming it into a different form, like the identity matrix during matrix inversion. Each row operation is associated with an elementary matrix, which characterizes the effect of the operation.

Using row operations, any invertible matrix can be expressed as a product of elementary matrices, ensuring a structured approach to matrix manipulation. During matrix inversion or expressing a matrix as a product of elementary matrices, understanding these operations becomes crucial.
Identity Matrix
The identity matrix, often denoted as \( I \), is the matrix's version of the number 1 in arithmetic. It plays a pivotal role in matrix algebra. An identity matrix is a square matrix (same number of rows and columns) with 1's along the main diagonal and 0's elsewhere. For a 3x3 matrix, it looks like this:
\[I = \left(\begin{array}{lll}1 & 0 & 0 \0 & 1 & 0 \0 & 0 & 1\end{array}\right)\]

The identity matrix is unique because it leaves any matrix unchanged when multiplied with it, similar to how multiplying a number by one keeps the number the same. This fundamental property makes the identity matrix particularly useful during matrix inversion and solving matrix equations. In operations or transformations, the goal is often to transform a matrix into the identity matrix for simplification.
  • The identity matrix is integral in defining the concept of an inverse matrix.
  • In any dimensional space, the identity matrix indicates no change or transformation.

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

Prove that if \(B\) is a \(3 \times 1\) matrix and \(C\) is a \(1 \times 3\) matrix, then the \(3 \times 3\) matrix \(B C\) has rank at most 1 . Conversely, show that if \(A\) is any \(3 \times 3\) matrix having rank 1 , then there exist a \(3 \times 1\) matrix \(B\) and a \(1 \times 3\) matrix \(C\) such that \(A=B C\).

Let $$ B=\left(\begin{array}{c|ccc} 1 & 0 & \cdots & 0 \\ \hline 0 & & & \\ \vdots & & B^{\prime} & \\ 0 & & & \end{array}\right) $$ where \(B^{\prime}\) is an \(m \times n\) submatrix of \(B\). Prove that if \(\operatorname{rank}(B)=r\), then \(\operatorname{rank}\left(B^{\prime}\right)=r-1 .\)

Let the reduced row echelon form of \(A\) be $$ \left(\begin{array}{rrrrr} 1 & 0 & 2 & 0 & -2 \\ 0 & 1 & -5 & 0 & -3 \\ 0 & 0 & 0 & 1 & 6 \end{array}\right) $$ Determine \(A\) if the first, second, and fourth columns of \(A\) are $$ \left(\begin{array}{r} 1 \\ -1 \\ 3 \end{array}\right), \quad\left(\begin{array}{r} 0 \\ -1 \\ 1 \end{array}\right), \quad \text { and } \quad\left(\begin{array}{r} 1 \\ -2 \\ 0 \end{array}\right) $$ respectively.Let the reduced row echelon form of \(A\) be $$ \left(\begin{array}{rrrrr} 1 & 0 & 2 & 0 & -2 \\ 0 & 1 & -5 & 0 & -3 \\ 0 & 0 & 0 & 1 & 6 \end{array}\right) $$ Determine \(A\) if the first, second, and fourth columns of \(A\) are $$ \left(\begin{array}{r} 1 \\ -1 \\ 3 \end{array}\right), \quad\left(\begin{array}{r} 0 \\ -1 \\ 1 \end{array}\right), \quad \text { and } \quad\left(\begin{array}{r} 1 \\ -2 \\ 0 \end{array}\right) $$ respectively.Let the reduced row echelon form of \(A\) be $$ \left(\begin{array}{rrrrr} 1 & 0 & 2 & 0 & -2 \\ 0 & 1 & -5 & 0 & -3 \\ 0 & 0 & 0 & 1 & 6 \end{array}\right) $$ Determine \(A\) if the first, second, and fourth columns of \(A\) are $$ \left(\begin{array}{r} 1 \\ -1 \\ 3 \end{array}\right), \quad\left(\begin{array}{r} 0 \\ -1 \\ 1 \end{array}\right), \quad \text { and } \quad\left(\begin{array}{r} 1 \\ -2 \\ 0 \end{array}\right) $$ respectively.Let the reduced row echelon form of \(A\) be $$ \left(\begin{array}{rrrrr} 1 & 0 & 2 & 0 & -2 \\ 0 & 1 & -5 & 0 & -3 \\ 0 & 0 & 0 & 1 & 6 \end{array}\right) $$ Determine \(A\) if the first, second, and fourth columns of \(A\) are $$ \left(\begin{array}{r} 1 \\ -1 \\ 3 \end{array}\right), \quad\left(\begin{array}{r} 0 \\ -1 \\ 1 \end{array}\right), \quad \text { and } \quad\left(\begin{array}{r} 1 \\ -2 \\ 0 \end{array}\right) $$ respectively.

Let \(A\) be an \(m \times n\) matrix and \(B\) be an \(n \times p\) matrix. Prove that \(A B\) can be written as a sum of \(n\) matrices of rank at most one.

It can be shown that the vectors \(u_{1}=(2,-3,1), u_{2}=(1,4,-2), u_{3}=\) \((-8,12,-4), u_{4}=(1,37,-17)\), and \(u_{5}=(-3,-5,8)\) generate \(\mathrm{R}^{3}\). Find a subset of \(\left\\{u_{1}, u_{2}, u_{3}, u_{4}, u_{5}\right\\}\) that is a basis for \(\mathrm{R}^{3}\).
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Decision 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.