Elementary Row & Column Operations

Topic 5 - Matrices

What are Elementary Operations?

Elementary operations are simple changes we can perform on the rows or columns of a matrix. They are used to simplify matrices, solve systems of equations, and find inverse matrices.

Types of Operations

Interchange: Swap two rows or two columns
Scaling: Multiply a row or column by a non-zero number
Replacement: Replace a row/column by adding multiple of another

1. Row Interchange (R₁ ↔ R₂)

Swap two rows.

1 2
3 4

R₁ ↔ R₂

3 4
1 2

2. Scaling (R₁ → kR₁)

Multiply all elements of a row by a non-zero constant k.

1 2
3 4

R₁ → 2R₁

2 4
3 4

3. Replacement (R₂ → R₂ + kR₁)

Add a multiple of one row to another row.

1 2
3 4

R₂ → R₂ + (-3)R₁

1 2
0 -2

Row vs Column Operations

All three operations also apply to columns (C₁, C₂, etc.)
Row operations are mainly used in solving equations
Column operations are used in determinants (next chapter)

Why are they important?

Simplify matrices to easier form
Used in Gauss elimination method
Help to find inverse of matrix
Reduce matrix to row-echelon form