The step-by-step process for calculating the transpose of a matrix:

1. Write down the original matrix. Let’s call it A.
2. Identify the number of rows and columns in the matrix A. Let’s call the number of rows m and the number of columns n.
3. Create a new matrix, called the transpose of A. This matrix will have n rows and m columns.
4. Copy the elements of the original matrix A to the new matrix in such a way that the rows of the original matrix become the columns of the
5. new matrix and the columns of the original matrix become the rows of the new matrix. That is, the element in the i-th row and j-th column of the original matrix should be placed in the j-th row and i-th column of the new matrix.
6. The resulting matrix is the transpose of the original matrix.
   For example, suppose we have the following matrix A:

A = [ [1, 2, 3],
[4, 5, 6]
]
To find the transpose of this matrix, we follow the steps:

Write down the original matrix A:

A = [ [1, 2, 3],
[4, 5, 6]
]
Identify the number of rows and columns in A:

m = 2, n = 3
Create a new matrix with swapped dimensions:

transpose = [
[0, 0],
[0, 0],
[0, 0]
]
Copy the elements of the original matrix to the new matrix in such a way that the rows of the original matrix become the columns of the new matrix and the columns of the original matrix become the rows of the new matrix:

transpose = [
[1, 4],
[2, 5],
[3, 6]
]
The resulting matrix is the transpose of the original matrix:

transpose of A = [ [1, 4],
[2, 5],
[3, 6]
]
So the transpose of the matrix A is [[1, 4], [2, 5], [3, 6]].