Python – Matrix

A matrix is a way to organize numbers in a rectangular grid made up of rows and columns. We can assume it like a table, where:

• Rows go across (left to right)
• Columns go down (top to bottom)

The size of a matrix is defined by the number of rows (m) and columns (n). If a matrix has 3 rows and 4 columns, it’s called a 3×4 matrix.

Matrices are used in:

• Solving linear equations
• Image transformations (rotations, scaling)
• Machine learning algorithms
• Data representation

In this tutorial, we’ll explore different ways to create and work with matrices in Python, including using the NumPy library for matrix operations.

Visual representation of a matrix

Creating a simple matrix using Python

Method 1: Creating a matrix with a List of list

A Matrix is fundamentally a 2D list therefore we can create a Matrix by creating a 2D list (list of lists).

mat = [[1, 2, 3, 4], 
    [5, 6, 7, 8],
    [9, 10, 11, 12]]

print("Matrix =", mat) 

Matrix = [[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12]]

Method 2: Take Matrix input from user in Python

In this example we are going to take user inputs for rows and columns for the matrix and then print the complete matrix.

rows = int(input("rows: "))
col = int(input("columns: "))

matrix = [] 
print("entries row-wise:")

for i in range(rows):   
    row = []
    for j in range(col):
        row.append(int(input()))    # user input for rows
    matrix.append(row)  # adding rows to the matrix

print("\n2D matrix is:")

for i in range(rows):
    for j in range(col):
        print(matrix[i][j], end=" ")
    print()

Output:

Enter the number of rows: 2
Enter the number of columns: 2
Enter the entries row-wise:
1
2
3
4

The 2D matrix is:
1 2 
3 4 

Method 3: Create a matrix using list comprehension

List comprehension is an elegant way to define and create a list in Python, we are using the range function for printing 4 rows and 4 columns.

matrix = [[col for col in range(4)] for row in range(4)]
print(matrix)

[[0, 1, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3]]

Explanation:

• Outer loop (for row in range(4)) runs 4 times to create 4 rows.
• Inner loop (for col in range(4)) fills each row with values 0 to 3.

Assigning Value in a matrix

Method 1: Assign value to an individual cell in Matrix

Here we are replacing and assigning value to an individual cell (1 row and 1 column = 11) in the Matrix.

x = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
x[1][1] = 11

print(x)

[[1, 2, 3], [4, 11, 6], [7, 8, 9]]

Method 2: Using Negative Indexing

We are assigning a value to an individual cell using negative indexing in this example (-2 row and -1 column = 21) in the Matrix.

x = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]

x[-2][-1] = 21  # row = -2 , column = -1
print(x)

Output:

[[1, 2, 3], [4, 5, 21], [7, 8, 9]]

Accessing Value in a matrix

Method 1: Direct Indexing

We can access elements of a Matrix by using its row and column index.

print("Element at (1,3):", x[0][2])
print("Element at (3,3):", x[2][2])

Output:

Element at (1,3): 3
Element at (3,3): 9

Method 2: Negative Indexing

Here, we are accessing elements of a Matrix by passing its row and column on negative indexing.

print(x[-1][-2]) 

Output:

8

Mathematical Operations with Matrix in Python

Example 1: Addition Using Loops

Let’s see how we can add two matrices using for-loop in Python.

x = [[1, 2, 3],[4, 5, 6],[7, 8, 9]]
y = [[9, 8, 7],[6, 5, 4],[3, 2, 1]]
res = [[0]*3 for _ in range(3)]

for i in range(len(x)):
    for j in range(len(x[0])):
        res[i][j] = x[i][j] + y[i][j]

for r in res:
    print(r)

Output:

[10, 10, 10]
[10, 10, 10]
[10, 10, 10]

Example 2: Addition & Subtraction with List Comprehension

Performing the Basic addition and subtraction using list comprehension.

x = [[1, 2, 3],
    [4, 5, 6],
    [7, 8, 9]]

y = [[9, 8, 7],
    [6, 5, 4],
    [3, 2, 1]]

# Matrix addition
add_res = [[x[i][j] + y[i][j] for j in range(len(x[0]))] for i in range(len(x))]

# Matrix subtraction
sub_res = [[x[i][j] - y[i][j] for j in range(len(x[0]))] for i in range(len(x))]


print("Matrix Addition:")
for row in add_res:
    print(row)

print("\nMatrix Subtraction:")
for row in sub_res:
    print(row)

Output:

Matrix Addition
[10, 10, 10]
[10, 10, 10]
[10, 10, 10]


Matrix Subtraction
[-8, -6, -4]
[-2, 0, 2]
[4, 6, 8]

Example 3: Python program to multiply and divide two matrices

Performing the basic multiplication and division of two matrices using Python loop.

x = [[2, 4, 6],
    [8, 10, 12],
    [14, 16, 18]]

y = [[1, 2, 3],
    [4, 5, 6],
    [7, 8, 9]]

# Element-wise multiplication
mult_res = [[x[i][j] * y[i][j] for j in range(3)] for i in range(3)]

# Element-wise integer division
div_res = [[x[i][j] // y[i][j] for j in range(3)] for i in range(3)]

print("Matrix Multiplication:")
for row in mult_res:
    print(row)

print("\nMatrix Division:")
for row in div_res:
    print(row)

Output:

Matrix Multiplication:
[2, 8, 18]
[32, 50, 72]
[98, 128, 162]

Matrix Division:
[2, 2, 2]
[2, 2, 2]
[2, 2, 2]

Transpose of a Matrix

Example 1: Using loop

Transpose of a matrix is obtained by changing rows to columns and columns to rows. In other words, transpose of A[][] is obtained by changing A[i][j] to A[j][i].

x = [[9, 8, 7], [6, 5, 4], [3, 2, 1]]
transpose = [[0]*3 for _ in range(3)]

for i in range(len(x)):
    for j in range(len(x[0])):
        transpose[j][i] = x[i][j]

for r in transpose:
    print(r)

[9, 6, 3]
[8, 5, 2]
[7, 4, 1]

Example 2: Using List Comprehension

Here’s how to Transpose a matrix using list comprehension.

transpose = [[x[j][i] for j in range(len(x))] for i in range(len(x[0]))]
for row in transpose:
    print(row)

Output:

[9, 6, 3]
[8, 5, 2]
[7, 4, 1]

NumPy Matrix Operations

Creating a Matrix with Random Values

Here we are creating a Numpy array using numpy.random and a random module.

import numpy as np

arr = np.random.randint(10, size=(3, 3))
print(arr)

Output:

[[2 7 5]
 [8 5 1]
 [8 4 6]]

Explanation:

• The numpy.random module is used to generate random numbers.
• np.random.randint(10, size=(3, 3)) creates a 3×3 matrix with random integers from 0 to 9.

Basic Math Operations with NumPy

Here we are covering different mathematical operations such as addition, subtraction, multiplication, and division using Numpy.

import numpy as np

x = np.array([[1, 2], [4, 5]])
y = np.array([[7, 8], [9, 10]])

print("Addition:\n", np.add(x, y))
print("Subtraction:\n", np.subtract(x, y))
print("Multiplication:\n", np.multiply(x, y))
print("Division:\n", np.divide(x, y))

Addition:
 [[ 8 10]
 [13 15]]
Subtraction:
 [[-6 -6]
 [-5 -5]]
Multiplication:
 [[ 7 16]
 [36 50]]
Division:
 [[0.14285714 0.25      ]
 [0.44444444 0.5       ]]

Dot and cross product with Matrix

In this example, we are going to discuss how we can calculate the dot and the cross products of two matrices using NumPy, it provides built in functions to calculate them. First, let’s discsuss what are dot and cross products in short:

• Dot Product: Calculates the sum of the products of corresponding elements, often used to find projections or perform matrix multiplication.
• Cross Product: Produces a vector perpendicular to two 3D vectors, useful in physics for torque, angular momentum, etc.

import numpy as np

x = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
y = [[9, 8, 7], [6, 5, 4], [3, 2, 1]]

print("Dot Product:\n", np.dot(x, y))
print("Cross Product:\n", np.cross(x, y))

Dot Product:
 [[ 30  24  18]
 [ 84  69  54]
 [138 114  90]]
Cross Product:
 [[-10  20 -10]
 [-10  20 -10]
 [-10  20 -10]]

Transpose Using NumPy

To perform transpose operation in matrix we can use the numpy.transpose() method.

import numpy as np

matrix = [[1, 2, 3], [4, 5, 6]]
print(np.transpose(matrix))

[[1 4]
 [2 5]
 [3 6]]

Creating Empty Matrices

Initializing an empty array, using the np.zeros().

import numpy as np

a = np.zeros((2, 2), dtype=int)
print("2x2 Matrix:\n", a)

b = np.zeros((3, 3))
print("3x3 Matrix:\n", b)

2x2 Matrix:
 [[0 0]
 [0 0]]
3x3 Matrix:
 [[0. 0. 0.]
 [0. 0. 0.]
 [0. 0. 0.]]

Slicing in Matrix using Numpy

Slicing is the process of choosing specific rows and columns from a matrix and then creating a new matrix by removing all of the non-selected elements.

import numpy as np

x = np.array([[6, 8, 10], 
              [9, -12, 15],
              [12, 16, 20],
              [15, -20, 25]])

print("Full Matrix:\n", x)
print("Third Row, Second Column:", x[2:3, 1])
print("Third Row, Third Column:", x[2:3, 2])

Full Matrix:
 [[  6   8  10]
 [  9 -12  15]
 [ 12  16  20]
 [ 15 -20  25]]
Third Row, Second Column: [16]
Third Row, Third Column: [20]

Deleting Rows with NumPy

Here, we are trying to delete rows using the np.delete() function. In the code, we first tried to delete the 0_th row, then we tried to delete the 2_nd row, and then the 3_rd row.

import numpy as np

a = np.array([[6, 8, 10], 
              [9, -12, 15],
              [12, 16, 20],
              [15, -20, 25]])

print("Delete 0th Row:\n", np.delete(a, 0, axis=0))
print("Delete 1st Row:\n", np.delete(a, 1, axis=0))

Delete 0th Row:
 [[  9 -12  15]
 [ 12  16  20]
 [ 15 -20  25]]
Delete 1st Row:
 [[  6   8  10]
 [ 12  16  20]
 [ 15 -20  25]]

Add Rows or Columns to a NumPy Array

We can easily add a new row or column to an existing NumPy array using stacking functions like np.hstack() (horizontal stack) and np.vstack() (vertical stack).

Here’s how to add a column to an existing 2D array:

import numpy as np

x = np.array([[6, 8, 10],
              [9, -12, 15],
              [15, -20, 25]])

# new column to be added
col = np.array([1, 2, 3])

# add the column (after reshaping it into a column vector)
res = np.hstack((x, np.atleast_2d(col).T))

print("Resultant Array:\n", res)

Resultant Array:
 [[  6   8  10   1]
 [  9 -12  15   2]
 [ 15 -20  25   3]]