Writing A Matrix Into Row Echelon Form

Definition: Row Echelon Form of a Matrix A matrix in Row Echelon Form (REF) has the following properties: 1. All rows consisting entirely of zeros occur at the bottom of the matrix. 2. *Below each leading coefficient (first non-zero entry in a given row, also referred to as a pivot) is a column of zeros. 3. Each pivot lies to the right of the pivot in the row above, producing a ”stair case” pattern. * In some books, they require that the leading coefficient of each non-zero row is 1. Example: Write the following matrix in row echelon form: 

0 −3 −6 4  −1 −2 −1 3   −2 −3 0 3 1 4 5 −9

 9 1   −1  −7

R1 ¿ R4 1 4 5  −1 −2 −1   −2 −3 0 0 −3 −6

−9 3 3 4



−7 1   −1  9

1  0   −2 0

4 2 −3 −3

5 4 0 −6

−9 −6 3 4

 −7 −6   −1  9

1 4  0 2   0 5 0 −3

5 4 10 −6

−9 −6 −15 4

 −7 −6   9  0

5 4 0 −6

   

In certain books, the above matrix is in Row Echelon Form. We can carry out one more step to obtain 1’s in the pivot position.

⇓ 2R1 + R3 → R3 

⇓ R3 ¿ R4  1 4 5 −9  0 2 4 −6   0 −3 −6 4 0 0 0 0

4 2 0 −3

⇓ (3/2)R2 + R3 → R3  1 4 5 −9 −7  0 2 4 −6 −6   0 0 0 −5 0 0 0 0 0 0

⇓ R1 + R2 → R2 

 −7 −6   0  9

1  0   0 0

⇓ 

−9 −6 0 4



⇓ (1/2)R2 → R2 (−1/5)R3 → R3



−7 −6   −15  9



1  0   0 0

⇓ (−5/2)R2 + R3 → R3

4 1 0 0

 5 −9 −7 2 −3 −3   0 1 0  0 0 0

The above matrix is in Row Echelon Form. The boxed numbers are the pivots.

1