Let the coefficient matrix be in reduced echelon form with a pivot in each column, since each matrix is equivalent to one and only one reduced echelon matrix. Construct a matrix with the dimensions determined in the previous step that is in reduced echelon form and has a pivot in each column.

Constructed matrix

0 1 2

1 2 1 A

2 7 8

⇒

1 2 1

0 1 2 A1

2 7 8

⇒

1 2 1

0 1 2 A2

0 3 6

Step-by-step explanation:

Echelon Forms

A matrix is in row echelon form (ref) when it satisfies the following conditions.

1. The first non-zero element in each row, called the leading entry, is 1.

2. Each leading entry is in a column to the right of the leading entry in the previous row.

3. Rows with all zero elements, if any, are below rows having a non-zero element.

A matrix is in reduced row echelon form (rref) when it satisfies the following conditions.

The matrix is in row echelon form (i. e., it satisfies the three conditions listed above).

The leading entry in each row is the only non-zero entry in its column.

A matrix in echelon form is called an echelon matrix. Matrix A and matrix B are examples of echelon matrices.

1 2 3 4

0 0 1 3

0 0 0 1

0 0 0 0

A

1 2 0 0

0 0 1 0

0 0 0 1

0 0 0 0

B

Matrix A is in row echelon form, and matrix B is in reduced row echelon form.

How to Transform a Matrix Into Its Echelon Forms

Any matrix can be transformed into its echelon forms, using a series of elementary row operations. Here's how.

1. Pivot the matrix

Find the pivot, the first non-zero entry in the first column of the matrix.

2. Interchange rows, moving the pivot row to the first row.

Multiply each element in the pivot row by the inverse of the pivot, so the pivot equals 1.

3. Add multiples of the pivot row to each of the lower rows, so every element in the pivot column of the lower rows equals 0.

4. To get the matrix in row echelon form, repeat the pivot

Repeat the procedure from Step 1 above, ignoring previous pivot rows.

5. Continue until there are no more pivots to be processed.

To get the matrix in reduced row echelon form, process non-zero entries above each pivot.

Identify the last row having a pivot equal to 1, and let this be the pivot row.

Add multiples of the pivot row to each of the upper rows, until every element above the pivot equals 0.

Moving up the matrix, repeat this process for each row.

To illustrate the transformation process, let's transform Matrix A to a row echelon form and to a reduced row echelon form.

0 1 2

1 2 1 A

2 7 8

⇒

1 2 1

0 1 2 A1

2 7 8

⇒

1 2 1

0 1 2 A2

0 3 6