np.linalg.inv() for matrix inverse in NumPy - Mini Project: Build & Apply

Calculate Matrix Inverse Using np.linalg.inv()

📖 Scenario: Imagine you are working with a small business that needs to solve a system of linear equations to optimize their resources. One way to do this is by finding the inverse of a matrix representing their data.

🎯 Goal: You will create a matrix using NumPy, set up a configuration variable, calculate the inverse of the matrix using np.linalg.inv(), and then print the result.

📋 What You'll Learn

Create a 2x2 NumPy array called matrix with values [[4, 7], [2, 6]]

Create a variable called identity_check to store the product of matrix and its inverse

Use np.linalg.inv() to calculate the inverse of matrix and store it in inverse_matrix

Print the inverse_matrix and identity_check to verify the inverse calculation

💡 Why This Matters

🌍 Real World

Matrix inversion is used in many fields like engineering, physics, computer graphics, and economics to solve systems of equations and model real-world problems.

💼 Career

Understanding matrix operations and using NumPy for linear algebra is essential for data scientists, machine learning engineers, and researchers working with numerical data.

1

Create the matrix

Create a 2x2 NumPy array called matrix with the exact values [[4, 7], [2, 6]].

NumPy

import numpy as np
# Create a 2x2 NumPy array called matrix with values [[4, 7], [2, 6]]
# Your code here

Hint

Use np.array() to create the matrix with the given values.

2

Set up identity check variable

Create a variable called identity_check and set it to None for now. This will later store the product of the matrix and its inverse.

NumPy

import numpy as np
matrix = np.array([[4, 7], [2, 6]])
# Create a variable called identity_check and set it to None
# Your code here

Hint

Initialize identity_check with None as a placeholder.

3

Calculate the inverse matrix

Use np.linalg.inv() to calculate the inverse of matrix and store it in a variable called inverse_matrix. Then calculate the product of matrix and inverse_matrix and store it in identity_check.

NumPy

import numpy as np
matrix = np.array([[4, 7], [2, 6]])
identity_check = None
# Calculate the inverse of matrix using np.linalg.inv() and store in inverse_matrix
# Calculate the product of matrix and inverse_matrix and store in identity_check
# Your code here

Hint

Use np.linalg.inv(matrix) to get the inverse. Use np.dot() to multiply matrices.

4

Print the inverse and identity check

Print the variables inverse_matrix and identity_check to see the inverse matrix and verify the product is close to the identity matrix.

NumPy

import numpy as np
matrix = np.array([[4, 7], [2, 6]])
identity_check = None
inverse_matrix = np.linalg.inv(matrix)
identity_check = np.dot(matrix, inverse_matrix)
# Print inverse_matrix and identity_check
# Your code here

Hint

Use two print() statements, one for inverse_matrix and one for identity_check.

Practice

1. What does the function np.linalg.inv() do in numpy?

easy

A. It calculates the inverse of a square matrix.

B. It computes the determinant of a matrix.

C. It transposes a matrix.

D. It multiplies two matrices.

Solution

1. Step 1: Understand the function purpose

   np.linalg.inv() is designed to find the inverse of a matrix, which is a matrix that when multiplied with the original gives the identity matrix.

2. Step 2: Differentiate from other matrix operations

   Calculating determinant, transpose, or multiplication are different functions in numpy, not np.linalg.inv().

3. Final Answer:

   It calculates the inverse of a square matrix. -> Option A

4. Quick Check:

   np.linalg.inv() = inverse matrix [OK]

Hint: Inverse matrix means undo multiplication [OK]

Common Mistakes:

• Confusing inverse with transpose
• Thinking it calculates determinant
• Assuming it multiplies matrices

2. Which of the following is the correct syntax to find the inverse of a matrix A using numpy?

easy

A. np.inv(A)

B. np.linalg.inverse(A)

C. np.linalg.inv(A)

D. np.inverse(A)

Solution

1. Step 1: Recall numpy linear algebra module

   The inverse function is inside the linalg module of numpy, so it must be called as np.linalg.inv().

2. Step 2: Check function names

   Functions like np.inv(), np.inverse(), or np.linalg.inverse() do not exist in numpy.

3. Final Answer:

   np.linalg.inv(A) -> Option C

4. Quick Check:

   Correct syntax = np.linalg.inv(A) [OK]

Hint: Use np.linalg.inv() for matrix inverse [OK]

Common Mistakes:

• Omitting 'linalg' module
• Using wrong function names
• Confusing with np.inverse()

3. You have a matrix C = np.array([[2, 3], [1, 4]]). You want to verify that np.linalg.inv(C) is correct by multiplying C with its inverse. What should the result be?

easy

A. An identity matrix of the same size as C.

B. The original matrix C itself.

C. A zero matrix of the same size as C.

D. A matrix with all elements equal to 1.

Solution

1. Step 1: Recall property of inverse matrices

   Multiplying a matrix by its inverse results in the identity matrix, which has 1s on the diagonal and 0s elsewhere.

2. Step 2: Apply to matrix C

   So, C @ np.linalg.inv(C) should produce the identity matrix of size 2x2.

3. Final Answer:

   An identity matrix of the same size as C. -> Option A

4. Quick Check:

   Matrix x inverse = identity matrix [OK]

Hint: Matrix times inverse equals identity [OK]

Common Mistakes:

• Expecting zero matrix instead
• Confusing with original matrix
• Thinking result is all ones

4. What is the output of the following code?

import numpy as np
A = np.array([[1, 2], [3, 4]])
inv_A = np.linalg.inv(A)
print(np.round(inv_A, 2))

medium

A. [[ 4. -2. ]
[-3. 1.5]]

B. [[ 1.5 -0.5 ]
[-2. 1. ]]

C. [[-2. 1. ]
[ 1.5 -0.5]]

D. [[ 0.5 -0.25]
[-0.75 0.25]]

Solution

1. Step 1: Calculate determinant of A

   Determinant = (1*4) - (2*3) = 4 - 6 = -2.

2. Step 2: Compute inverse using formula

   Inverse = (1/det) * [[4, -2], [-3, 1]] = (-1/2) * [[4, -2], [-3, 1]] = [[-2, 1], [1.5, -0.5]].

3. Step 3: Check numpy output

   Numpy returns approximately [[-2., 1.], [1.5, -0.5]]. np.round(inv_A, 2) prints as [[-2. 1. ] [ 1.5 -0.5]], matching [[-2. 1. ] [ 1.5 -0.5]].

4. Final Answer:

   [[ 0.5 -0.25] [-0.75 0.25]] -> Option D

5. Quick Check:

   np.linalg.inv(A) rounded = [[ 0.5 -0.25] [-0.75 0.25]] [OK]

Hint: Inverse = adjoint / determinant [OK]

Common Mistakes:

• Confusing determinant sign
• Not rounding output
• Mixing up matrix elements

5. What is the error in the following code snippet?

import numpy as np
B = np.array([[1, 2, 3], [4, 5, 6]])
inv_B = np.linalg.inv(B)
print(inv_B)

medium

A. TypeError because np.linalg.inv() expects a list.

B. ValueError because B is not a square matrix.

C. SyntaxError due to missing parentheses.

D. No error; code runs fine.

Solution

1. Step 1: Check matrix shape

   Matrix B has shape (2, 3), which is not square (rows != columns).

2. Step 2: Understand np.linalg.inv() requirements

   np.linalg.inv() requires a square matrix to compute the inverse. Non-square matrices cannot have inverses.

3. Final Answer:

   ValueError because B is not a square matrix. -> Option B

4. Quick Check:

   Non-square matrix = ValueError [OK]

Hint: Inverse only for square matrices [OK]

Common Mistakes:

• Trying inverse on non-square matrix
• Confusing error type
• Assuming code runs without error