np.linalg.solve() for linear systems in NumPy - Cheat Sheet & Quick Revision

Recall & Review

beginner

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

np.linalg.solve() finds the solution to a system of linear equations Ax = b, where A is a matrix and b is a vector or matrix of constants.

beginner

What are the required inputs for np.linalg.solve(A, b)?

You need a square matrix A (same number of rows and columns) and a vector or matrix b representing the constants on the right side of the equations.

intermediate

Why must matrix A be square for np.linalg.solve()?

Because only square matrices have a unique inverse, which is needed to find a unique solution to the system Ax = b.

intermediate

What happens if matrix A is singular or not invertible when using np.linalg.solve()?

np.linalg.solve() will raise a LinAlgError because the system does not have a unique solution.

beginner

How can you interpret the output of np.linalg.solve(A, b)?

The output is the vector or matrix x that satisfies the equation Ax = b. It gives the values of the variables that solve the system.

What shape must matrix A have to use np.linalg.solve(A, b)?

A Rectangular matrix

B Only 1D array

C Any shape

D Square matrix

Matrix A must be square (same number of rows and columns) for np.linalg.solve() to work.

What does np.linalg.solve() return?

A The inverse of matrix <code>A</code>

B The solution vector or matrix <code>x</code>

C The determinant of <code>A</code>

D The product of <code>A</code> and <code>b</code>

It returns the solution x that satisfies Ax = b.

If A is singular, what will happen when calling np.linalg.solve(A, b)?

A It returns zero vector

B It returns a random vector

C It raises a <code>LinAlgError</code>

D It returns <code>b</code>

A singular matrix has no unique inverse, so np.linalg.solve() raises an error.

Which equation does np.linalg.solve(A, b) solve?

A <code>Ax = b</code>

B <code>xA = b</code>

C <code>Ab = x</code>

D <code>b = Ax + 1</code>

The function solves the linear system Ax = b.

What type of data can b be in np.linalg.solve(A, b)?

A A vector or matrix

B Only a vector

C Only a scalar

D Only a matrix

b can be a vector or a matrix representing multiple right-hand sides.

Explain how np.linalg.solve() is used to solve a system of linear equations.

Think about the equation form and what the function returns.

What errors or issues might you encounter when using np.linalg.solve() and how can you handle them?

Consider what happens if the system has no 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