Row reduction and echelon forms  Linear Equations with Matrices
Row reduction and echelon forms
Related Concepts
 Solving a linear system with matrices using Gaussian elimination
Lessons
Notes:
A nonzero row is a row that has at least one entry that is not zero.
The leading entry of a row is the leftmost nonzero entry in a nonzero row.
A rectangular matrix is in echelon form if it has the three properties:
 All nonzero rows are above any rows of all zeros.
 Each leading entry of a row is in a column to right of the leading entry of the row above it.
 All entries in a column below a leading entry are zeros.
If the rectangular matrix satisfies 2 more additional properties, then it is in reduced echelon form:
 The leading entry in each nonzero row is 1.
 Each leading 1 is the only nonzero entry in its column.
Essentially the difference between the two forms is that the reduced echelon form has 1 as a leading entry, and the column of the leading entry has 0's below and above.
The pivot position is just the leading entries of the echelon form matrix.
The pivot column is the column of the pivot position.
Here are the steps for the row reduction algorithm:
 Look for the leftmost nonzero column. This is our pivot column.
 Find a nonzero entry in the pivot column. This is our pivot position. It should be at the very top of the pivot column.
 Use matrix row operations to make all the entries below the pivot 0.
 Ignore the row with the pivot. Repeat Step 13 again and again until you can't anymore.
 Find your rightmost pivot. Make it 1. Then make all the entries above it 0.

Intro Lesson
Row Reduction and Echelon Form Overview:

1.
Difference between Echelon Form and Reduced Echelon Form
Label whether the following matrices are in echelon form or reduced echelon form: 
2.
Learning the Row Reduction Algorithm
Row reduce the matrices to reduced echelon form. Circle the pivot positions in the final and original matrix, and list the pivot columns from the original matrix in part b: 
3.
Finding the General Solution of a Matrix
Find the general solution of the following matrices: 
4.
Linear Systems with Unknown Constants
Choose values of a and b where the system has infinitely many solutions, and no solutions: