Echelon Forms, Pivots & Pivot Columns (Linear Algebra)

0:00 Useful Views of Augmented Matrices
1:08 Echelon Form & Leading Entries
3:56 Reduced Echelon Form & Pivot Positions
5:01 Uniqueness of Reduced Echelon Form

We explain what it means for a matrix to be in echelon form and reduced echelon form in this video from our linear algebra series. We begin by looking at examples of systems of linear equations and the augmented matrices that represent them. We explore how having zero entries in certain places in a matrix can help us more easily determine the solution for a system of equations.

We introduce leading entries in a matrix as the first nonzero entry in row, and we discuss the requirement of the locations of leading entries in an echelon form of a matrix. Not all rows will have a leading entry. All entries below a leading entry in a column must be zero. We then explain the requirement for a matrix to be in reduced echelon form. Each leading entry of 1 in reduced echelon form is called a pivot position, and each column containing a pivot position is referred to as a pivot column.

Finally, we mention how the reduced echelon form of a matrix is unique, but the echelon form of a matrix is not unique.