1. Linear Equations in Linear Algebra

1.2 Row Reduction and Echelon Forms

DefinitionA rectangular matrix is in echelon form (or row echelon form)if it has the following three properties:

1. All nonzero rows are above any rows of all zeros.2. Each leading entry of a row is in a column to the right of the leading entry of the row above it.3. All entries in a column below a leading entry are zeros.

If a matrix in echelon form satisfies the following two conditions,then it is in reduced echelon form (or reduced row echelon form):

4. The leading entry in each nonzero row is 1.5. Each leading 1 is the only nonzero entry in its column.

Leading entry: leftmost nonzero entry

Example1

*000000

**00000

****000

******0

Reduced Echelon form

Echelon form

*1000000

*0100000

*00*1000

*00*0*10

5.000

8310

1232

5100

1010

3001

Examples:

Reduced echelon form ?Echelon form?

5.003

8310

1232

5100

1010

3101

Echelon form? Reduced echelon form?

Echelon form?

Echelon form?

Reduced echelon form?

Reduced echelon form?

Theorem (Uniqueness of the Reduced Echelon Form)

Each matrix is row equivalent to one and only one reduced echelon matrix.

DefinitionA pivot position in a matrix A is a location in A that corresponds to a leading 1 in the reduced echelon form of A.

A pivot column is a column of A that contains a pivot position.

11000

20100

30001

AofREF

102102

162104

295306

A

Pivot position

Pivot column

Note: When row operations produce a matrix in echelon form, further row operations to obtain the reduced echelon form do not change the position of the leading entries.

Elementary Row OperationsUsing the TI83

Matrix/math

• B: rref(matrix)

• C: rowSwap(matrix, rowA, rowB)

• D: row+(matrix, rowA, rowB)

• E: *row(value, matrix, row)

• F:*row+(value, matrix, rowA, rowB)

1654

1223

22

62

Solve

qzyx

qzyx

qyx

qzyx

1

5

2

1

q

z

y

x