Algebra 54 - Gaussian Elimination | Summary and Q&A

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Algebra 54 - Gaussian Elimination

TL;DR

Gaussian elimination is a method used to simplify augmented matrices, specifically to transform them into row echelon form. Matrices in row echelon form have leading entries of 1, zeros below the leading entries, and each leading entry is to the right of the leading entry above it.

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Questions & Answers

Q: What is row echelon form?

Row echelon form is a simplified form of an augmented matrix where the leading entry in each row is 1, all entries to the right of the leading entry can have any value, and each leading entry is to the right of the leading entry above it. Rows that contain all zeros are positioned at the bottom of the matrix.

Q: What are the elementary row operations used in Gaussian elimination?

The three elementary row operations used in Gaussian elimination are swap, scale, and pivot. Swap involves exchanging rows, scale involves multiplying a row by a constant, and pivot involves adding or subtracting a multiple of one row to another row.

Q: How does back substitution work?

Back substitution is a process used to solve a system of linear equations once the augmented matrix is in row echelon form. It involves working backwards from the last equation to eliminate variables until each equation contains only a single variable. The solutions can then be obtained by plugging in the values of the solved variables into the original equations.

Q: What is the difference between Gaussian elimination and Gauss-Jordan elimination?

Gaussian elimination transforms the augmented matrix into row echelon form, while Gauss-Jordan elimination further simplifies the matrix to reduced row echelon form. Reduced row echelon form has the additional requirement that the leading entry in each row is the only non-zero entry in its column.

Summary & Key Takeaways

  • Gaussian elimination is a process used to simplify augmented matrices into row echelon form.

  • Matrices in row echelon form have specific requirements, including leading entries of 1, zeros below the leading entries, and each leading entry is to the right of the leading entry above it.

  • The process of Gaussian elimination involves performing elementary row operations to transform the matrix, such as swapping rows, scaling rows, and pivoting rows.

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