Transpose of a Matrix

What is Transpose?

Transpose of a matrix is obtained by interchanging its rows and columns. That means rows become columns and columns become rows.

Example

Given Matrix A:

Transpose Aᵀ:

Rows of A become columns in Aᵀ.

Step-by-Step Explanation

• First row (1 2 3) becomes first column
• Second row (4 5 6) becomes second column
• Resulting matrix has order 3 × 2

Important Properties

• (Aᵀ)ᵀ = A
• (A + B)ᵀ = Aᵀ + Bᵀ
• (AB)ᵀ = BᵀAᵀ
• (kA)ᵀ = kAᵀ