why is the inverse of matrix the transpose code example

Example: what is the transpose of a matrix

The transpose of a matrix is just a flipped version of 
the original matrix. We can transpose a matrix by switching 
its rows with its columns. The original rows become the new columns
and the original columns become the new rows.
We denote the transpose of matrix A by AT. 

For example:
	1 2 3                      1 4 7
A = 4 5 6       then      AT = 2 5 8
    7 8 9                      3 6 9
    
Similarly if 

B = 1 2 3       then      BT = 1 4
    4 5 6                      2 5 
                               3 6