Class Notes: SM-7 Reduced Row Echelon Form
Goal: Solve systems of equations with matrices
x − 2y + z = 7
3 x − 5 y + z = 14 is equivalent to
2x − 2 y − z = 3
⎡1 −2 1 7 ⎤
⎢ 3 −5 1 14 ⎥
⎢
⎥
⎢⎣ 2 −2 −1 3 ⎥⎦
x − 2y + z = 7
⎡ 1 −2 1 7 ⎤
⎢ 0 1 −2 −7 ⎥ is equivalent to
y − 2 z = −7 Row Echelon Form
⎢
⎥
⎢⎣ 0 0 1
3 ⎥⎦
z =3
⎡1 0 0 2 ⎤
⎢ 0 1 0 −1⎥ is equivalent to
⎢
⎥
⎢⎣ 0 0 1 3 ⎥⎦
x=2
y = −1 This would be ideal – Reduced Row Echelon Form.
z =3
Reduced Row Echelon Form is unique.
⎡ 1 −2 1 7 ⎤
⎢ 3 −5 1 14 ⎥
⎢
⎥
⎢⎣ 2 −2 −1 3 ⎥⎦
Find the reduced row echelon form for each system. Write the solution to the system.
Example 1:
x − 2 y + z = −2
2x − 3y + 2z = 2
4 x − 8 y + 5 z = −5
Example 2:
x + 3z = 5
y − 2 z = −2
Interpreting Reduced Row Echelon Form
Write the solution to the following systems
⎡1 −3 1 4 ⎤
⎢ 0 −1 −4 7 ⎥
⎢
⎥
⎢⎣ 0 0 0 5 ⎥⎦
⎡1
⎢0
⎢
⎢0
⎢
⎣0
0 1 2 1⎤
1 −2 −1 −1⎥⎥
0 0 0 0⎥
⎥
0 0 0 0⎦