Matrix multiplication with @ operator in NumPy - Cheat Sheet & Quick Revision

Recall & Review

beginner

What does the @ operator do in numpy?

The @ operator performs matrix multiplication between two numpy arrays. It multiplies rows of the first matrix by columns of the second matrix and sums the products.

beginner

How is matrix multiplication different from element-wise multiplication?

Matrix multiplication combines rows and columns to produce a new matrix, while element-wise multiplication multiplies corresponding elements directly without combining rows and columns.

intermediate

Given two matrices A of shape (2, 3) and B of shape (3, 4), what will be the shape of A @ B?

The result will have shape (2, 4) because the inner dimensions (3) match and the output shape is the outer dimensions (2 from A and 4 from B).

intermediate

What error occurs if you try to multiply matrices with incompatible shapes using @?

You get a ValueError saying shapes are not aligned because the number of columns in the first matrix must equal the number of rows in the second matrix.

beginner

Write a simple numpy code snippet to multiply two matrices A and B using the @ operator.

import numpy as np
A = np.array([[1, 2], [3, 4]])
B = np.array([[5, 6], [7, 8]])
result = A @ B
print(result)

What does the @ operator do in numpy?

A Adds two matrices

B Performs element-wise multiplication

C Subtracts two matrices

D Performs matrix multiplication

The @ operator is used for matrix multiplication in numpy.

If matrix A has shape (3, 2) and matrix B has shape (2, 4), what is the shape of A @ B?

A (2, 2)

B (3, 4)

C (3, 2)

D (4, 3)

The output shape is (3, 4) because the inner dimensions 2 match and outer dimensions are 3 and 4.

What happens if you try to multiply matrices with shapes (2, 3) and (2, 4) using @?

A It works and returns shape (2, 4)

B It returns a scalar

C It raises a ValueError due to shape mismatch

D It performs element-wise multiplication

Matrix multiplication requires the number of columns in the first matrix to equal the number of rows in the second matrix.

Which numpy function is equivalent to using the @ operator?

A np.dot()

B np.multiply()

C np.add()

D np.cross()

np.dot() performs matrix multiplication, same as the @ operator.

What is the result of multiplying a (2, 2) identity matrix with another (2, 2) matrix using @?

A The other matrix unchanged

B A zero matrix

C A matrix of ones

D An error

Multiplying by the identity matrix returns the other matrix unchanged.

Explain how the @ operator works for matrix multiplication in numpy.

Think about how you multiply matrices by hand.

Describe what happens if you try to multiply two numpy arrays with incompatible shapes using the @ operator.

Consider the rules for matrix multiplication dimensions.

Practice

1. What does the @ operator do in numpy when applied between two arrays?

easy

A. Performs matrix multiplication if shapes are compatible

B. Adds the two arrays element-wise

C. Calculates the element-wise product

D. Computes the transpose of the first array

Solution

1. Step 1: Understand the @ operator purpose

   The @ operator in numpy is designed for matrix multiplication, which requires the inner dimensions of the two arrays to match.

2. Step 2: Differentiate from other operations

   Element-wise addition or multiplication use + or * respectively, not @. Transpose uses .T.

3. Final Answer:

   Performs matrix multiplication if shapes are compatible -> Option A

4. Quick Check:

   @ means matrix multiply [OK]

Hint: Remember: @ means matrix multiply, not element-wise [OK]

Common Mistakes:

• Confusing @ with element-wise multiplication
• Thinking @ adds arrays
• Assuming @ transposes arrays

2. Which of the following is the correct syntax to multiply two numpy arrays A and B using the @ operator?

easy

A. C = A * B

B. C = A + B

C. C = A.dot(B)

D. C = A @ B

Solution

1. Step 1: Identify the @ operator usage

   The @ operator is used as C = A @ B to perform matrix multiplication in numpy.

2. Step 2: Differentiate from other operations

   A * B is element-wise multiplication, A.dot(B) is a method but not using @, and A + B is addition.

3. Final Answer:

   C = A @ B -> Option D

4. Quick Check:

   Use @ between arrays for matrix multiply [OK]

Hint: Use @ directly between arrays for matrix multiply [OK]

Common Mistakes:

• Using * instead of @ for matrix multiply
• Confusing method dot() with operator @
• Using addition operator + mistakenly

3. What is the output of the following code?

import numpy as np
A = np.array([[1, 2], [3, 4]])
B = np.array([[5, 6], [7, 8]])
C = A @ B
print(C)

medium

A. [[ 5 12]
[21 32]]

B. [[ 6 8]
[10 12]]

C. [[19 22]
[43 50]]

D. [[ 5 6]
[ 7 8]]

Solution

1. Step 1: Calculate matrix multiplication manually

   Multiply rows of A by columns of B:
   First row: (1*5 + 2*7) = 19, (1*6 + 2*8) = 22
   Second row: (3*5 + 4*7) = 43, (3*6 + 4*8) = 50

2. Step 2: Confirm output matches calculation

   The resulting matrix is [[19, 22], [43, 50]], which matches [[19 22] [43 50]].

3. Final Answer:

   [[19 22] [43 50]] -> Option C

4. Quick Check:

   Matrix multiply result = [[19 22] [43 50]] [OK]

Hint: Multiply rows by columns and sum for each element [OK]

Common Mistakes:

• Adding elements instead of multiplying and summing
• Mixing element-wise multiplication with matrix multiplication
• Confusing row and column order

4. What error will occur when running this code?

import numpy as np
A = np.array([[1, 2, 3], [4, 5, 6]])
B = np.array([[7, 8], [9, 10]])
C = A @ B

medium

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

B. TypeError: unsupported operand type(s) for @

C. No error, output is a (2,2) matrix

D. IndexError: index out of bounds

Solution

1. Step 1: Check shapes of arrays

   Array A shape is (2,3), array B shape is (2,2). For matrix multiplication, A's columns (3) must equal B's rows (2).

2. Step 2: Identify mismatch and error

   Since 3 != 2, numpy raises a ValueError about shape misalignment.

3. Final Answer:

   ValueError: shapes (2,3) and (2,2) not aligned for matrix multiplication -> Option A

4. Quick Check:

   Matrix multiply needs matching inner dimensions [OK]

Hint: Check inner dimensions match before using @ [OK]

Common Mistakes:

• Ignoring shape mismatch and expecting output
• Confusing element-wise multiplication with matrix multiplication
• Assuming @ works like addition

5. Given two numpy arrays:

A = np.array([[1, 0], [0, 1]])
B = np.array([[2, 3], [4, 5]])

What is the result of C = A @ B @ A?

hard

A. [[5 8]
[9 14]]

B. [[2 3]
[4 5]]

C. [[1 0]
[0 1]]

D. [[2 4]
[3 5]]

Solution

1. Step 1: Multiply A and B

   Matrix A is the identity matrix. Multiplying identity with B returns B:
   A @ B = B = [[2, 3], [4, 5]]

2. Step 2: Multiply result by A again

   Multiplying B by identity matrix A again returns B:
   B @ A = B = [[2, 3], [4, 5]]

3. Final Answer:

   [[2 3] [4 5]] -> Option B

4. Quick Check:

   Identity matrix leaves other matrix unchanged [OK]

Hint: Identity matrix A leaves matrix unchanged when multiplied [OK]

Common Mistakes:

• Multiplying incorrectly and swapping rows/columns
• Assuming multiplication changes matrix when identity is involved
• Confusing element-wise and matrix multiplication