How to Tell if a Matrix is in EF, REF, or RREF – And Solve the System
Title: How to Tell if a Matrix is in EF, REF, or RREF – And Solve the System
This post explains how to tell the difference between Echelon Form (EF), Row-Echelon Form (REF), and Reduced Row-Echelon Form (RREF), using a simple matrix example and then solving the associated system of equations step by step.
Given Matrix:
A =
[1 0 8 0]
[0 1 5 -1]
[0 0 0 0]
Key Definitions:
-
EF (Echelon Form) – Has a staircase pattern of leading entries with zeros below.
-
REF (Row-Echelon Form) – Same as EF but with leading 1s in each pivot row.
-
RREF (Reduced Row-Echelon Form) – Same as REF, but each leading 1 has zeros above and below it.
Answer: The matrix is in EF, REF, and RREF.
Dimensions, Rank, and Size:
-
Dimensions (size) = 3 rows × 4 columns → 3x4
-
Rank = number of pivot positions = 2
-
Dimension of A = 2
Solving the System
Assuming the matrix is an augmented system, separate the last column as the constants vector.
Coefficient matrix:
[1 0 8]
[0 1 5]
[0 0 0]
Constants vector:
[ 0 ]
[-1 ]
[ 0 ]
Turn into Equations:
-
x + 8z = 0 → x = -8z
-
y + 5z = -1 → y = -1 - 5z
-
z = z (free variable)
Final Answers:
-
General solution:
x = -8z, y = -1 - 5z, z = free -
Vector solution:
x = [ 0 ] + z [ -8 ]
[ -1 ] [ -5 ]
[ 0 ] [ 1 ] -
Parametric form:
x(t) = (-8t, -5t - 1, t)
This breakdown comes from Chapter 7: Matrices and Determinants in the Ultimate Crash Course for STEM Majors, built to help you confidently solve systems and understand matrix structure clearly.
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