Transpose of a Matrix

In linear algebra, the transpose of a matrix is an operation that flips the matrix over its diagonal, switching the row and column indices of each element. For a given matrix , the transpose is denoted by .

Mathematically, if

then the transpose is given by

Example 1: Transpose of a 2x2 Matrix

Consider the 2x2 matrix

The transpose of , denoted by , is obtained by swapping its rows and columns:

Example 2: Transpose of a 3x3 Matrix

Now consider the 3x3 matrix

The transpose of , denoted by , is:

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