In Python, we can implement a matrix as nested list (list inside a list).

We can treat each element as a row of the matrix.

For example `X = [[1, 2], [4, 5], [3, 6]]`

would represent a `3x2`

matrix.

The first row can be selected as `X[0]`

. And, the element in first row, first column can be selected as `X[0][0]`

.

Multiplication of two matrices `X` and `Y` is defined only if the number of columns in `X` is equal to the number of rows `Y`.

If `X` is a `n x m`

matrix and `Y` is a `m x l`

matrix then, `XY` is defined and has the dimension `n x l`

(but `YX` is not defined). Here are a couple of ways to implement matrix multiplication in Python.

### Source Code: Matrix Multiplication using Nested Loop

```
# Program to multiply two matrices using nested loops
# 3x3 matrix
X = [[12,7,3],
[4 ,5,6],
[7 ,8,9]]
# 3x4 matrix
Y = [[5,8,1,2],
[6,7,3,0],
[4,5,9,1]]
# result is 3x4
result = [[0,0,0,0],
[0,0,0,0],
[0,0,0,0]]
# iterate through rows of X
for i in range(len(X)):
# iterate through columns of Y
for j in range(len(Y[0])):
# iterate through rows of Y
for k in range(len(Y)):
result[i][j] += X[i][k] * Y[k][j]
for r in result:
print(r)
```

**Output**

[114, 160, 60, 27] [74, 97, 73, 14] [119, 157, 112, 23]

In this program, we have used nested `for`

loops to iterate through each row and each column. We accumulate the sum of products in the result.

This technique is simple but computationally expensive as we increase the order of the matrix.

For larger matrix operations we recommend optimized software packages like NumPy which is several (in the order of 1000) times faster than the above code.

### Source Code: Matrix Multiplication Using Nested List Comprehension

```
# Program to multiply two matrices using list comprehension
# 3x3 matrix
X = [[12,7,3],
[4 ,5,6],
[7 ,8,9]]
# 3x4 matrix
Y = [[5,8,1,2],
[6,7,3,0],
[4,5,9,1]]
# result is 3x4
result = [[sum(a*b for a,b in zip(X_row,Y_col)) for Y_col in zip(*Y)] for X_row in X]
for r in result:
print(r)
```

The output of this program is the same as above. To understand the above code we must first know about built-in function `zip()`

and unpacking argument list using * operator.

We have used nested list comprehension to iterate through each element in the matrix. The code looks complicated and unreadable at first. But once you get the hang of list comprehensions, you will probably not go back to nested loops.