np.linalg.solve() for linear systems in NumPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of solving a 2x2 linear system

What is the output of the following code that solves a system of linear equations?

import numpy as np
A = np.array([[3, 1], [1, 2]])
b = np.array([9, 8])
x = np.linalg.solve(A, b)
print(x)

NumPy

import numpy as np
A = np.array([[3, 1], [1, 2]])
b = np.array([9, 8])
x = np.linalg.solve(A, b)
print(x)

A [3. 2.]

B [2. 3.]

C [1. 4.]

D [4. 1.]

💡 Hint

Remember that np.linalg.solve finds x in Ax = b.

The system is 3x + y = 9 and x + 2y = 8. Solving gives x=2, y=3.

❓ data_output

intermediate

Number of solutions for a singular matrix

What happens when you try to solve a linear system with a singular matrix using np.linalg.solve()? Consider this code:

import numpy as np
A = np.array([[1, 2], [2, 4]])
b = np.array([5, 10])
x = np.linalg.solve(A, b)

What is the result?

NumPy

import numpy as np
A = np.array([[1, 2], [2, 4]])
b = np.array([5, 10])
x = np.linalg.solve(A, b)

A Returns solution [5, 10]

B Returns solution [1, 2]

C Returns solution [0, 0]

D Raises a LinAlgError due to singular matrix

💡 Hint

Singular matrices do not have unique solutions.

The matrix A is singular because the second row is a multiple of the first, so np.linalg.solve raises a LinAlgError.

🔧 Debug

advanced

Identify the error in solving a linear system

What error will this code produce?

import numpy as np
A = np.array([[1, 2, 3], [4, 5, 6]])
b = np.array([7, 8])
x = np.linalg.solve(A, b)

NumPy

import numpy as np
A = np.array([[1, 2, 3], [4, 5, 6]])
b = np.array([7, 8])
x = np.linalg.solve(A, b)

A LinAlgError: Last 2 dimensions of the array must be square

B ValueError: shapes (2,3) and (2,) not aligned

C TypeError: unsupported operand type(s) for +: 'int' and 'str'

D No error, returns solution

💡 Hint

np.linalg.solve requires a square coefficient matrix.

Matrix A is 2x3, not square, so np.linalg.solve raises LinAlgError about non-square matrix.

🚀 Application

advanced

Using np.linalg.solve to find intersection point

Two lines are given by equations:
3x + 2y = 13
7x - y = 19
Use np.linalg.solve to find the intersection point (x, y). What is the output?

NumPy

import numpy as np
A = np.array([[3, 2], [7, -1]])
b = np.array([13, 19])
x = np.linalg.solve(A, b)
print(x)

A [4. 1.]

B [2. 3.]

C [3. 2.]

D [1. 5.]

💡 Hint

Set up the system as Ax = b and solve.

Solving the system gives x=3 and y=2, which satisfies both equations.

🧠 Conceptual

expert

Understanding conditions for np.linalg.solve success

Which condition must be true for np.linalg.solve(A, b) to successfully find a unique solution to Ax = b?

A Matrix A must be square and invertible

B Matrix A must be rectangular with more rows than columns

C Vector b must be all positive values

D Matrix A must be symmetric

💡 Hint

Think about when a system has a unique solution.

np.linalg.solve requires A to be square and invertible (non-singular) to find a unique solution.