result = [] # final result
for i in range(len(A)):

    row = [] # the new row in new matrix
    for j in range(len(B[0])):
        
        product = 0 # the new element in the new row
        for v in range(len(A[i])):
            product += A[i][v] * B[v][j]
        row.append(product) # append sum of product into the new row
        
    result.append(row) # append the new row into the final result


print(result)

I have basically two comments:

1. Naming.
2. Use functions to follow the single responsibility principle.

Naming

Python has an official style-guide, PEP8. Some of its recommendations are:

• Using lower_case for variables and functions, PascalCase for classes and ALL_CAPS for global constants.
• To limit your line length to 80 (or at the most 120) characters

Your names which you use to iterate through the rows and columns are not clear at all. Why vc and vr (I guess the c and r stand for column and row, but what does the v stand for?)? Instead I would use rows and columns for the number of rows and columns and use i and j to iterate over them, because that is the convention used in mathematics when describing matrices.

Single responsibility principle

As for your actual code, you basically have four major responsibilities:

1. Asking the user for input
2. Creating a matrix
3. Calculating the matrix product
4. Printing a matrix scheme

I would make this into four functions:

def ask_user(message):
    while True:
        try:
            return int(input(message))
        except TypeError:
            pass

This function will ask the user the message until he inputs something that can be cast to an int.

def matrix_from_user():
    rows = ask_user("Number of rows?: ")
    columns = ask_user("Number of columns?: ")
    matrix = [[0] * columns for _ in range(rows)]
    for i in range(rows):
        for j in range(columns):
            matrix[i][j] = ask_user("M{}{}: ".format(i+1, j+1))
    return matrix

Here I used str.format to make the construction of the string easier.

def matrix_mul(matrix1, matrix2):
    rows, columns = len(matrix1), len(matrix2[0])
    matrix = [[0] * columns for _ in range(rows)]
    for i in range(rows):
        for j in range(columns):
            matrix[i][j] = sum(matrix1[i][k] * matrix2[k][j]
                               for k in range(len(matrix2)))
    return matrix

This is a straightforward implementation of the formula \ for two matrices \ and \. It uses the built-in sum function that takes an iterable to sum. This iterable is a generator comprehension here.

def matrix_print(matrix):
    print('\nElements of matrix:')
    rows, columns = len(matrix), len(matrix[0])
    for i in range(rows):
        row = ' '.join('M{}{}'.format(i + 1, j + 1) for j in range(columns))
        print('|', row, '|')

Here I used str.join to easily join an iterable of strings with a given delimiter (' ' here).

Your main program becomes then simply:

if __name__ == "__main__":
    matrix1 = matrix_from_user()
    matrix_print(matrix1)
    matrix2 = matrix_from_user()

    product = matrix_mul(matrix1, matrix2)

    print('Matrix1: ', matrix1)
    print('Matrix2: ', matrix2)
    print('Result of Matrix Multiplication:', product)

This code I put under an if __name__ == "__main__": guard which allows you to import this script from another script without this part of the code being run (so you get access to the functions).

Hello the variable k you are using will save the results in position 0 and 1 in first loop and it stores the second column results also in same position 0 and 1. So you won't get the final result of first columns except the last one. Instead you can initialize an empty array with values 0 and you can add the values in their respective positions.

mat1 = [[3,4],[2,1]

mat2 = [[1,5],[3,7]]   

result = [[0, 0],[0, 0]]

for i in range(len(mat1)):

    for j in range(len(mat2[0])):

        for k in range(len(mat2)):

            result[i][j] += mat1[i][k] * mat2[k][j]

for r in result:
    print(r)

You could use a list (or a dictionary) to hold your matrices:

matrices = []

for i in range(len(node2)):

    if (sOrP[i] == 'S'):
        #print('series connection')
        matrixTemplate = numpy.array([[1.0, 0.0],[0.0, 1.0]]) #Got to put 1.0 else it doesnt work

        matrixTemplate[0][1] = frequenciesList[0][i]

        matrices.append(matrixTemplate)

        #print(matrixTemplate)

    elif (sOrP[i] == 'P'):
        #print('parallel connection')
        matrixTemplate = numpy.array([[1.0, 0.0],[0.0, 1.0]])

        matrixTemplate[1][0] = 1/frequenciesList[0][i]

        matrices.append(matrixTemplate)

        #print(matrixTemplate)

And then use reduce and numpy.matmul to compute your total matrix product:

from functools import reduce

Ty = reduce(numpy.matmul, matrices)