20 RREF Practice Problems

Complete with Matrix Solutions

15 min read
By Illia Spencer

Table of Contents

Transform these matrices into row reduced echelon form. Get step-by-step help with free RREF calculator.

Beginner Level (2×2 Matrices)

Problem 1

[2 4]
[6 8]

Answer:
[1 2]
[0 0]

Problem 2

[1 2]
[3 4]

Answer:
[1 0]
[0 1]

Problem 3

[5 10]
[3 6]

Answer:
[1 2]
[0 0]

Problem 4

[2 3]
[4 6]

Answer:
[1 1.5]
[0 0]

Problem 5

[1 -2]
[3 -6]

Answer:
[1 -2]
[0 0]

Need verification? Check your work here

Intermediate Level (3×3 Matrices)

Problem 6

[1 2 3]
[4 5 6]
[7 8 9]

Answer:
[1 0 -1]
[0 1 2]
[0 0 0]

Problem 7

[1 0 2]
[0 1 3]
[2 1 7]

Answer:
[1 0 0]
[0 1 0]
[0 0 1]

Problem 8

[2 0 4]
[0 2 6]
[0 0 2]

Answer:
[1 0 0]
[0 1 0]
[0 0 1]

Problem 9

[1 2 0]
[0 1 0]
[0 0 1]

Answer:
[1 0 0]
[0 1 0]
[0 0 1]

Problem 10

[3 6 9]
[2 4 6]
[1 2 3]

Answer:
[1 2 3]
[0 0 0]
[0 0 0]

Advanced Level (Augmented & 4×4 Matrices)

Problem 11

[1 2 3 | 4]
[2 3 4 | 5]
[3 4 5 | 6]

Answer:
[1 0 -1 | 2]
[0 1 2 | -1]
[0 0 0 | 0]

Problem 12

[1 2 3 4]
[0 1 2 3]
[0 0 1 2]
[0 0 0 1]

Answer:
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]

Problem 13

[1 -1 2 | 3]
[2 -2 4 | 6]
[3 -3 6 | 9]

Answer:
[1 -1 2 | 3]
[0 0 0 | 0]
[0 0 0 | 0]

Problem 14

[1 2 3 4]
[5 6 7 8]
[9 10 11 12]
[13 14 15 16]

Answer:
[1 0 -1 -2]
[0 1 2 3]
[0 0 0 0]
[0 0 0 0]

Problem 15

[1 1 1 | 6]
[1 2 3 | 14]
[1 3 4 | 18]

Answer:
[1 0 0 | 2]
[0 1 0 | 4]
[0 0 1 | 0]

Problem 16

[2 4 | 6]
[3 6 | 9]

Answer:
[1 2 | 3]
[0 0 | 0]

Problem 17

[1 2 3 4]
[2 4 6 8]
[3 6 9 12]

Answer:
[1 2 3 4]
[0 0 0 0]
[0 0 0 0]

Problem 18

[1 0 0 | 2]
[0 1 0 | 3]
[0 0 1 | 4]

Answer:
[1 0 0 | 2]
[0 1 0 | 3]
[0 0 1 | 4]

Problem 19

[1 2 | 3]
[4 5 | 6]

Answer:
[1 0 | -1]
[0 1 | 2]

Problem 20

[1 -1 2 3]
[2 -2 4 6]
[3 -3 6 9]

Answer:
[1 -1 2 3]
[0 0 0 0]
[0 0 0 0]

Quick Tips

  1. Start with leftmost nonzero column
  2. Make each pivot 1
  3. Create zeros above/below pivots
  4. Verify with RREF Calculator