np.linalg.inv() for matrix inverse in NumPy - Cheat Sheet & Quick Revision

Recall & Review

beginner

What does np.linalg.inv() do in NumPy?

np.linalg.inv() calculates the inverse of a square matrix. The inverse matrix, when multiplied by the original, gives the identity matrix.

beginner

What kind of matrix can np.linalg.inv() invert?

Only square matrices (same number of rows and columns) that are non-singular (have a non-zero determinant) can be inverted using np.linalg.inv().

intermediate

What happens if you try to invert a singular matrix with np.linalg.inv()?

NumPy will raise a LinAlgError because singular matrices do not have an inverse.

beginner

How can you verify that a matrix A and its inverse A_inv are correct?

Multiply A by A_inv. The result should be the identity matrix, which has 1s on the diagonal and 0s elsewhere.

beginner

Write a simple example to compute the inverse of a 2x2 matrix using np.linalg.inv().

<pre>import numpy as np
A = np.array([[4, 7], [2, 6]])
A_inv = np.linalg.inv(A)
print(A_inv)</pre>

What type of matrix can np.linalg.inv() invert?

A Only singular matrices

B Any rectangular matrix

C Square and non-singular matrices

D Only diagonal matrices

Only square matrices with a non-zero determinant (non-singular) can be inverted.

What error does np.linalg.inv() raise if the matrix is singular?

A ValueError

B LinAlgError

C TypeError

D IndexError

NumPy raises a LinAlgError when trying to invert a singular matrix.

What is the result of multiplying a matrix by its inverse?

A Zero matrix

B Transpose matrix

C Diagonal matrix

D Identity matrix

The product of a matrix and its inverse is the identity matrix.

Which NumPy module contains the inv() function?

A numpy.linalg

B numpy.random

C numpy.fft

D numpy.core

The inv() function is in the numpy.linalg module.

What shape must a matrix have to be invertible?

A Square (same rows and columns)

B Rectangular with more rows than columns

C Rectangular with more columns than rows

D Any shape

Only square matrices can have an inverse.

Explain in your own words what np.linalg.inv() does and when you can use it.

Think about what an inverse matrix means in real life.

Describe how you would check if a matrix inverse calculation was successful using NumPy.

What is the identity matrix?

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