A good questions! Here, I used the swap formula from Numpy. When I tried to use one dimension for [b] in this code, like b[i] = b[k] or b[[i]] = b[[k]] , it gave me totally wrong results. When I used the same formula for [a], it worked perfectly, as you see in the video. I think it works in a similar way to the swap formula by using Python tuples, like a , b = b , a , but for indexes i and k. Because Numpy uses internal algorithms for such techniques, I had to make some trials and compare them with exact solutions until I got the best working formula. I hope I could answer your question.

You can input a matrix or a system (a-matrix & b-vector) in different ways. The simplest way is by using the Python input() function, so the user can define the matrix during the run of the code. The disadvantage of this method is that it is very tedious and error-prone to enter the matrices and vectors manually at each run, especially when they are relatively large (like 10x10 or more). The more practical method is to write the system in a text file, then open and read the file from the Python code. It is recommended for numeric data (numbers) to use CSV (Comma Separated Values) format to avoid reading errors. Since Python has tools to read from an Excel file, this could be another choice if you use Excel to save the systems. I will make a video about this topic soon. I really appreciate your question. Thank you.

@@mechtutorcom thanks for replying...🙇🙇.I shall definitely try it.Hey...do you not think that you should extend the number of your python tutorials of numerical analysis?I like the way you explain and deserve more related videos.Thank you.

Elements of Numpy arrays, like Python lists, can be accessed by slicing. Slicing uses built-in pre-compiled codes to return or modify elements in the arrays according to their indices. a[[k,i]]=a[[i,k]] is an application of this technique. Both k and i refer to rows not columns. k is the index of the current pivot row in the elimination process. i is index of the row that to be swapped with the k row if the condition is satisfied during the partial pivoting. The break command is used here to stop swapping of rows k and i if the condition is satisfied. If you remove the break command, the partial pivoting will continue to the last row, which will increase the run time of the code. The advantage of removing the break is that the pivot row (k) at the end will have the largest pivot element in its column. The indentation of the break statement should keep it inside the block of the condition. Otherwise, the i-loop will be exited in an uncontrolled way. Thanks for you comment.

your question helped me. thanks