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.

Practice

1. What does np.linalg.solve(A, b) do in NumPy?

easy

A. Solves the system of linear equations Ax = b for x

B. Calculates the determinant of matrix A

C. Finds the inverse of matrix A

D. Multiplies matrix A by vector b

Solution

1. Step 1: Understand the function purpose

   np.linalg.solve() is designed to find the vector x that satisfies the equation Ax = b, where A is a square matrix and b is a vector.

2. Step 2: Differentiate from other matrix operations

   Calculating determinant, inverse, or multiplication are different operations and use other functions like np.linalg.det(), np.linalg.inv(), or @ operator respectively.

3. Final Answer:

   Solves the system of linear equations Ax = b for x -> Option A

4. Quick Check:

   np.linalg.solve() = solve Ax=b [OK]

Hint: It finds x in Ax = b, not determinant or inverse [OK]

Common Mistakes:

• Confusing solve() with matrix inverse
• Using solve() for non-square matrices
• Thinking it multiplies matrices

2. Which of the following is the correct syntax to solve the system Ax = b using NumPy?

easy

A. np.linalg.solve(b, A)

B. np.solve.linalg(A, b)

C. np.linalg.solve(A, b)

D. np.solve(A, b)

Solution

1. Step 1: Recall correct function call

   The correct function is np.linalg.solve() with the first argument as matrix A and second as vector b.

2. Step 2: Check argument order and module

   Arguments must be (A, b), not reversed. The function is inside np.linalg, not np.solve.

3. Final Answer:

   np.linalg.solve(A, b) -> Option C

4. Quick Check:

   Correct syntax = np.linalg.solve(A, b) [OK]

Hint: Remember: np.linalg.solve(matrix, vector) [OK]

Common Mistakes:

• Swapping A and b arguments
• Using wrong module or function name
• Missing np.linalg prefix

3. What is the output of this code?

import numpy as np
A = np.array([[2, 1], [1, 3]])
b = np.array([10, 15])
x = np.linalg.solve(A, b)
print(x)

medium

A. [2. 5.]

B. [4. 3.]

C. [5. 2.]

D. [3. 4.]

Solution

1. Step 1: Set up equations from matrix and vector

   Matrix A and vector b represent:
   2x + 1y = 10
   1x + 3y = 15

2. Step 2: Solve equations manually or trust np.linalg.solve

   Solving:
   From first: y = (10 - 2x)
   Substitute in second: x + 3(10 - 2x) = 15
   x + 30 - 6x = 15
   -5x = -15
   x = 3
   y = 10 - 2*3 = 4
   np.linalg.solve gives x = [3. 4.]

3. Final Answer:

   [3. 4.] -> Option D

4. Quick Check:

   np.linalg.solve(A,b) = [3. 4.] [OK]

Hint: Use np.linalg.solve to get exact solution vector [OK]

Common Mistakes:

• Mixing up order of variables in solution
• Incorrect manual calculation
• Confusing rows and columns in matrix

4. 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)

medium

A. ValueError: shapes (3,2) and (2,) not aligned

B. LinAlgError: Last 2 dimensions of the array must be square

C. TypeError: unsupported operand type(s)

D. No error, returns solution vector

Solution

1. Step 1: Check matrix shape requirements

   Matrix A must be square (same number of rows and columns) to use np.linalg.solve(). Here, A is 3x2, not square.

2. Step 2: Identify error raised by NumPy

   NumPy raises LinAlgError with message about last 2 dimensions needing to be square.

3. Final Answer:

   LinAlgError: Last 2 dimensions of the array must be square -> Option B

4. Quick Check:

   Non-square A causes LinAlgError [OK]

Hint: Matrix A must be square for np.linalg.solve() [OK]

Common Mistakes:

• Using non-square matrix A
• Expecting multiplication instead of solve
• Ignoring error message details

5. You have the system:

3x + 2y - z = 1
2x - 2y + 4z = -2
-x + 0.5y - z = 0

Which code correctly solves for x, y, z using np.linalg.solve()?

hard

A. A = np.array([[3, 2, -1], [2, -2, 4], [-1, 0.5, -1]])
b = np.array([1, -2, 0])
x = np.linalg.solve(A, b)

B. A = np.array([[3, 2, 1], [2, -2, 4], [-1, 0.5, -1]])
b = np.array([1, -2, 0])
x = np.linalg.solve(A, b)

C. A = np.array([[3, 2, -1], [2, -2, 4], [-1, 0.5, -1]])
b = np.array([1, 2, 0])
x = np.linalg.solve(A, b)

D. A = np.array([[3, 2, -1], [2, -2, 4], [-1, 0.5, -1]])
b = np.array([1, -2, 0])
x = np.linalg.inv(A) @ b

Solution

1. Step 1: Translate system to matrix and vector

   Coefficients matrix A must match the system exactly:
   Row 1: 3, 2, -1
   Row 2: 2, -2, 4
   Row 3: -1, 0.5, -1
   Constants vector b is [1, -2, 0].

2. Step 2: Check each option for correctness

   A = np.array([[3, 2, -1], [2, -2, 4], [-1, 0.5, -1]]) b = np.array([1, -2, 0]) x = np.linalg.solve(A, b) matches coefficients and constants exactly.
   A = np.array([[3, 2, 1], [2, -2, 4], [-1, 0.5, -1]]) b = np.array([1, -2, 0]) x = np.linalg.solve(A, b) has wrong sign in third element of first row.
   A = np.array([[3, 2, -1], [2, -2, 4], [-1, 0.5, -1]]) b = np.array([1, 2, 0]) x = np.linalg.solve(A, b) has wrong second element in b.
   A = np.array([[3, 2, -1], [2, -2, 4], [-1, 0.5, -1]]) b = np.array([1, -2, 0]) x = np.linalg.inv(A) @ b uses inverse multiplication which is less efficient and not recommended.

3. Final Answer:

   Option A code correctly solves the system -> Option A

4. Quick Check:

   Correct matrix and vector with np.linalg.solve() [OK]

Hint: Match coefficients and constants exactly; use np.linalg.solve() [OK]

Common Mistakes:

• Wrong signs in matrix or vector
• Using inverse instead of solve()
• Mixing up constants vector values