Matrix inverse (inv) in SciPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of matrix inverse calculation

What is the output of this code that calculates the inverse of a 2x2 matrix using scipy?

SciPy

import numpy as np
from scipy.linalg import inv

matrix = np.array([[4, 7], [2, 6]])
result = inv(matrix)
print(result)

A

[[ 1.5 -1.7]
 [-0.2  0.4]]

B

[[ 0.6 -0.7]
 [-0.2  0.4]]

C

[[ 0.6  0.7]
 [ 0.2  0.4]]

D

[[ 0.6 -0.7]
 [ 0.2 -0.4]]

💡 Hint

Recall the formula for inverse of 2x2 matrix [[a,b],[c,d]] is (1/(ad-bc))*[[d,-b],[-c,a]]

The determinant is (4*6 - 7*2) = 10. The inverse is (1/10)*[[6, -7], [-2, 4]] = [[0.6, -0.7], [-0.2, 0.4]].

❓ data_output

intermediate

Number of elements in inverse matrix

After computing the inverse of a 3x3 matrix using scipy.linalg.inv, how many elements does the resulting matrix contain?

SciPy

import numpy as np
from scipy.linalg import inv

matrix = np.array([[1,2,3],[0,1,4],[5,6,0]])
inverse = inv(matrix)
print(inverse.size)

A 9

B 6

C 3

D 1

💡 Hint

The inverse matrix has the same shape as the original matrix.

A 3x3 matrix has 9 elements. Its inverse also has 9 elements.

🔧 Debug

advanced

Error type when inverting a singular matrix

What error does this code raise when trying to invert a singular matrix using scipy.linalg.inv?

SciPy

import numpy as np
from scipy.linalg import inv

singular_matrix = np.array([[1, 2], [2, 4]])
inv(singular_matrix)

A IndexError: index out of bounds

B ValueError: Input must be square

C LinAlgError: Singular matrix

D TypeError: Unsupported operand type(s)

💡 Hint

A singular matrix has zero determinant and cannot be inverted.

The scipy.linalg.inv function raises a LinAlgError with message 'Singular matrix' when the matrix is not invertible.

🧠 Conceptual

advanced

Effect of matrix inverse on identity matrix

If you multiply a matrix by its inverse, what is the resulting matrix?

A A matrix with all elements equal to one

B A zero matrix

C The original matrix squared

D The identity matrix

💡 Hint

Think about what an inverse matrix means in multiplication.

Multiplying a matrix by its inverse returns the identity matrix, which acts like 1 in matrix multiplication.

🚀 Application

expert

Using matrix inverse to solve linear equations

Given the system of equations represented by Ax = b, where A is a 2x2 matrix and b is a vector, which code correctly computes x using scipy.linalg.inv?

SciPy

import numpy as np
from scipy.linalg import inv

A = np.array([[3, 1], [1, 2]])
b = np.array([9, 8])

A

x = inv(A).dot(b)
print(x)

B

x = b.dot(inv(A))
print(x)

C

x = inv(b).dot(A)
print(x)

D

x = A.dot(inv(b))
print(x)

💡 Hint

Remember the formula x = A^-1 * b for solving Ax = b.

To solve Ax = b, multiply the inverse of A by b. The dot product order matters.