A Matrix into Row Echelon Form, Reduced Row Echelon Form (Canonical Form), Normal Form, and Finding Rank

Notes on Matrix Transformations



A Matrix into Row Echelon Form, Reduced
Row Echelon Form (Canonical Form), Normal
Form, and Finding Rank




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, 1. Definitions
• Row Echelon Form (REF / Echelon Form):
Zero rows at the bottom. Pivots (first non-zero entry in each row) move to the right. All
entries below each pivot are zero.

• Reduced Row Echelon Form (RREF / Canonical Form):
All pivots are exactly 1, and each pivot column has zeros above and below the pivot.

• Normal Form (Smith Normal Form):
Diagonal form for integer matrices where each diagonal entry divides the next one.

• Rank:
Number of non-zero rows in REF or RREF.




2. Elementary Row Operations
1. Swap two rows

2. Multiply a row by a non-zero number

3. Add a multiple of one row to another row




3. General Procedure
For Echelon Form (REF):
Start from left column → Find pivot → Eliminate below → Move right.

For Canonical Form (RREF):
Get REF → Make pivots = 1 → Eliminate above pivots.

For Rank:
Count non-zero rows in RREF.




4. 10 Examples (Easy to Complex) with Full Step-by-Step Solutions




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