Gaussian Elimination and Row Echelon Form

Gaussian Elimination

• Gaussian elimination is a process of row reduction used to simplify augmented matrices.
• It involves using elementary row operations (swap, scale, pivot) to transform a matrix into row echelon form.

Row Echelon Form

• In row echelon form, the left-most non-zero entry in each row (the leading entry) must be 1.
• The entries to the right of the leading entry can be any value, zero or otherwise.
• Each row's leading entry must lie to the right of the leading entry in the row above it.
• Rows with all zeros must be positioned at the bottom of the matrix.
• Zeros below the leading entries typically form a stair-step pattern.
• No column contains more than one leading entry.

Gaussian Elimination Process

• Start with the augmented matrix representing a system of linear equations.
• Work from left to right and top to bottom, modifying entries in each column until the matrix is in row echelon form.
• Goal: Make each column contain no more than one leading entry, with leading entries being ones positioned to the right of leading entries in rows above.
• Use elementary row operations (swap, scale, pivot) to achieve this.

Working with Columns

• Column 1: Ensure the top entry is 1 (using swap, scale, or pivot). Zero out the entries below the leading entry using pivot operations.
• Column 2: Move on to the next column. Ensure the leading entry in the current row is 1 (using scale). Zero out the entry below the leading entry using a pivot operation.
• Repeat for subsequent columns: Continue moving right, ensuring each column contains only one leading entry, which is a 1.

Back Substitution

• After Gaussian elimination, the augmented matrix is in row echelon form.
• Convert the matrix back into equations, which are now simpler.
• Solve the equations backward: Starting with the last equation (which has only one variable), find its value.
• Substitute that value into the second-to-last equation, eliminating the variable.
• Continue substituting values into the remaining equations, working back to the first equation, solving for each variable.

Gauss-Jordan Elimination

• A second row reduction process that follows Gaussian elimination.
• Simplifies the augmented matrix even further.
• Allows for determining solutions directly from the matrix without converting it back into equations.

This quiz covers the concepts of Gaussian elimination and the characteristics of row echelon form. It will test your understanding of the processes involved in simplifying augmented matrices using elementary row operations.