1. What Is the Transpose of a Matrix?
The transpose of a matrix is obtained by interchanging its rows and columns. In other words, the first row becomes the first column, the second row becomes the second column, and so on.
The transpose of a matrix \( A \) is written as \( A^T \) or \( A' \).
Formally, if \( A = [a_{ij}] \), then its transpose is:
\( A^T = [a_{ji}] \)
2. How Transpose Rearranges a Matrix
To find the transpose:
- Write rows of the original matrix as columns.
- Write columns of the original matrix as rows.
The size of the matrix changes from m × n to n × m.
2.1. Example 1: Transpose of a 2 × 3 Matrix
Let:
\( A = \begin{bmatrix} 1 & 4 & 7 \\ 2 & 5 & 8 \end{bmatrix} \)
Rows become columns:
\( A^T = \begin{bmatrix} 1 & 2 \\ 4 & 5 \\ 7 & 8 \end{bmatrix} \)
Here:
- A is 2 × 3
- \( A^T \) is 3 × 2
2.2. Example 2: Transpose of a Square Matrix
Let:
\( B = \begin{bmatrix} 3 & -1 \\ 6 & 4 \end{bmatrix} \)
The transpose is:
\( B^T = \begin{bmatrix} 3 & 6 \\ -1 & 4 \end{bmatrix} \)
3. Important Facts About Transpose
Transpose has some simple but powerful properties:
- (Transpose of transpose) \( (A^T)^T = A \)
- (Transpose of sum) \( (A + B)^T = A^T + B^T \)
- (Transpose of scalar product) \( (kA)^T = kA^T \)
- (Transpose of product) \( (AB)^T = B^T A^T \) (order reverses)
5. Why Transpose Is Useful
The transpose operation is used in many areas of mathematics:
- Solving systems of equations
- Matrix multiplication involving vectors
- Defining symmetric and skew-symmetric matrices
- Working with orthogonal matrices
- Linear transformations and geometry
It is one of the most basic and frequently used matrix operations.