Gaussian Elimination Flashcards

Questions and Answers

What is an augmented matrix?

A matrix with no relation to linear equations

A matrix that only contains the constant terms

A matrix containing only the coefficients of a system of linear equations

A matrix that consists of the coefficients and the constant terms in a system of linear equations (correct)

What is Gaussian elimination?

The process of using row operations to transform a linear system into one whose augmented matrix is in row echelon form.

In row echelon form, the first nonzero entry in each nonzero row must be 0.

False

What does a row operation include?

All of the above Signup and view all the answers

What is the Gauss-Jordan Elimination Method?

Solving a system of linear equations by transforming an augmented matrix so that it is in reduced row-echelon form. Signup and view all the answers

What is an inverse matrix?

The matrix that when multiplied by the original matrix results in the identity matrix Signup and view all the answers

What is an identity matrix?

A square matrix that, when multiplied by another matrix, equals that same matrix. Signup and view all the answers

What is Reduced Row Echelon Form (RREF)?

An augmented matrix in which the first nonzero element of each row of the coefficient portion of the matrix is 1 and the rest of the elements in the same column as this element are 0. Signup and view all the answers

Study Notes

Matrix Terminology

Augmented Matrix: Combines coefficients and constant terms of a linear system into a single matrix format for easier manipulation.
Gaussian Elimination: A systematic method involving row operations that converts the augmented matrix of a linear system into row echelon form, allowing for easier solution finding.

Row Operations

Row Echelon Form (REF): A matrix is in REF when each leading entry (first non-zero number) of a row is 1, and is to the right of the leading entry of the previous row.
Row Operation Types: Include switching two rows, scaling a row by a non-zero constant, and adding one row to another to produce new row values.

Advanced Techniques

Gauss-Jordan Elimination Method: Further extends Gaussian elimination to achieve reduced row-echelon form (RREF), which simplifies solving linear systems thoroughly.
Reduced Row Echelon Form (RREF): A matrix where each leading entry is 1, and all entries above and below each leading 1 are 0, making back-substitution straightforward.

Key Matrix Concepts

Inverse Matrix: A matrix that, when multiplied with its original, yields the identity matrix, essential for solving matrix equations.
Identity Matrix: A special square matrix where all diagonal elements are 1 and all off-diagonal elements are 0, serving as the multiplicative identity in matrix algebra.

Description

Master the concepts of Gaussian elimination with these flashcards. Learn key terms such as augmented matrix and row echelon form essential for solving linear equations. Perfect for students looking to solidify their understanding of linear algebra techniques.