/*! 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 63 Using only 0 's and 1 's, list a... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 63

Using only 0 's and 1 's, list all possible \(2 \times 2\) matrices in row canonical form.

Short Answer

Expert verified
The possible \(2 \times 2\) matrices in row canonical form using only 0's and 1's are: \( \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}, \begin{bmatrix} 1 & 1 \\ 0 & 0 \end{bmatrix} , \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}, \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \)

Step by step solution

01

Determine the options for the first row

First, we need to determine what our options for the first row of the matrices are. We can have either: 1. a leading 1 in column 1: \( \begin{bmatrix} 1 & * \end{bmatrix} \) 2. a row of all zeros: \( \begin{bmatrix} 0 & 0 \end{bmatrix} \) where * represents a "to be determined" value.
02

Determine the options for the second row

Next, we will determine the options for the second row of the matrices: 1. If the first row has a leading 1 in the first column, the second row must have a 0 in the first column to be in row canonical form, giving two possibilities for the second-row entry in column 2: \( \begin{bmatrix} 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 1 \end{bmatrix} \) 2. If the first row is all zeros, the only possibility for the second row is also all zeros. Now, we can put all of this together to get the complete matrices in row canonical form.
03

Combine the rows

Finally, we will combine the choices for the first row and second row to get the final \(2 \times 2\) matrices in row canonical form: 1. If the first row has a leading 1 in the first column: a. With * = 0: \( \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} \) b. With * = 1: \( \begin{bmatrix} 1 & 1 \\ 0 & 0 \end{bmatrix} \), \( \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} \) 2. If the first row is all zeros: a. Only option: \( \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \) So, the possible \(2 \times 2\) matrices in row canonical form using only 0's and 1's are: \( \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}, \begin{bmatrix} 1 & 1 \\ 0 & 0 \end{bmatrix} , \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}, \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \)

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.

Matrices in Linear Algebra
Matrices are fundamental components in the field of linear algebra, serving as a compact way to represent and operate on sets of linear equations. A matrix is essentially a rectangular array of numbers, symbols, or expressions arranged in rows and columns. Linear algebra focuses on various operations involving matrices, such as addition, multiplication, and inversion, as well as finding their determinant and working with vector spaces.

One of the key processes in linear algebra is transforming a matrix into simpler forms to solve linear systems more easily. The row canonical form, also known as row echelon form, is an example where the matrix is simplified for solving linear equations. When a matrix is in this form, it has a staircase-like structure, where each succeeding row has a leading entry that is to the right of the leading entry in the previous row, and entries directly below leading coefficients are zeros.
Binary Matrices
Binary matrices are a specific type of matrix in which each element is either a 0 or a 1. These matrices find use in computer science, coding theory, graph theory, and various other applications where a binary decision-making process is required. Binary operations such as logical AND, OR, and XOR can be applied to binary matrices, and they differ from regular arithmetic operations.

In the context of the exercise, when constructing a binary matrix in row canonical form, the leading entry in any row can only be a 1 or the entire row can consist of 0's. This simplification helps identify all possible configurations that a binary matrix can take while still adhering to the strict structure of row canonical form. Binary matrices are essential in understanding structures that can be described in an on/off, yes/no, or true/false manner.
Reduced Row Echelon Form
Reduced row echelon form (RREF) is a refinement of the row canonical form that further simplifies the matrix to a state where it can be used to find the solutions to a system of linear equations more readily. Each leading 1 in a row is not only to the right of the leading 1 in the row above, but all entries in the column above and below the leading 1 are zeros.

Moreover, each leading 1 is the only non-zero entry in its column, making it a pivot position. This form offers a clear depiction of the relationships between variables in a linear system and is pivotal for certain calculations in linear algebra, such as determining the rank of a matrix or finding the inverse of invertible square matrices.

The exercise showcases how to build the reduced row echelon form for binary matrices, demonstrating that the options are highly limited when using only 0's and 1's, therefore rendering the systems they represent straightforward to analyze.

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 \(u_{1}=(1,2,4), u_{2}=(2,-3,1), u_{3}=(2,1,-1)\) in \(\mathbf{R}^{3} .\) Show that \(u_{1}, u_{2}, u_{3}\) are orthogonal, and write \(v\) as a linear combination of \(u_{1}, u_{2}, u_{3},\) where (a) \(v=(7,16,6),\) (b) \(v=(3,5,2)\)

Determine the pivot and free variables in each of the following systems: $$ \begin{array}{rrr} 2 x_{1}-3 x_{2}-6 x_{3}-5 x_{4}+2 x_{5}=7 & 2 x-6 y+7 z=1 & x+2 y-3 z=2 \\ x_{3}+3 x_{4}-7 x_{5}=6 & 4 y+3 z=8 & 2 x+3 y+z=4 \\ x_{4}-2 x_{5}=1 & 2 z=4 & 3 x+4 y+5 z=8 \end{array} $$ (a) (b) (c) (a) In echelon form, the leading unknowns are the pivot variables, and the others are the free variables. Here \(x_{1}\), \(x_{3}, x_{4}\) are the pivot variables, and \(x_{2}\) and \(x_{5}\) are the free variables.

Prove Theorem \(3.19: B\) is row equivalent to \(A\) (written \(B \sim A\) ) if and only if there exists a nonsingular matrix \(P\) such that \(B=P A\)

Describe the Gauss-Jordan elimination algorithm, which also row reduces an arbitrary matrix \(A\) to its row canonical form. The Gauss-Jordan algorithm is similar in some ways to the Gaussian elimination algorithm, except that here each pivot is used to place 0 's both below and above the pivot, not just below the pivot, before working with the next pivot. Also, one variation of the algorithm first normalizes each row-that is, obtains a unit pivot-before it is used to produce 0's in the other rows, rather than normalizing the rows at the end of the algorithm.

Using only 0 's and 1 's, find the number \(n\) of possible \(3 \times 3\) matrices in row canonical form.
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

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.