Triangular and Echelon Forms

1. Triangular Form
2. Echelon Forms
3. Existence/Uniqueness Theorem

1. Triangular Form

For any square matrix, the main diagonal are the entries where the row # = column #. A square matrix is called upper triangular if all the entries below the diagonal are 0, or lower triangular if all the entries above the diagonal are 0.

When a matrix is simplified to triangular form it becomes much easier to solve and back substitute. In order to turn a matrix into triangular form, follow the procedure:

Turn 1st column into all 0s (other than the top entry)
Turn 2nd column into all 0s (other than the top 2 entries)
Repeat for all columns

2. Echelon Forms

2.1. Row Echelon Form

The analogue to triangular form for non-square matrices is the row echelon form (REF). Matrices in REF satisfy the following two conditions:

All the zero rows are at the bottom
The leading entry of each row is to the right of the row above it

The leading entry of a row is the first number in that row that is nonzero.

Note that a key difference between REF and upper triangular form is that the upper triangular form makes no conditions on the leading entry of each row, so it is possible that a matrix is in upper triangular form but not in REF.

2.2. Reduced Row Echelon Form

A more simplified form of this is the reduced row echelon form, or RREF. Matrices in RREF satisfy all the conditions of REF, plus the following two conditions:

Each leading entry is 1
All columns with leading entries are all 0 other than the leading entry itself

An important property of the RREF is that every matrix has a unique RREF, which can be useful for certain proofs.

2.3. Pivots

A pivot position is a location in a matrix which has a leading 1 in its RREF. We can find the pivots by locating the leading entries in a matrix that is in REF.

3. Existence/Uniqueness Theorem

Given the augmented matrix of a system,

If a pivot is in the final column, then the system is not consistent
If there is no pivot in the final column, then it is consistent
If all prior columns have pivots, then there is a unique solution