Matrix multiplication with @ operator in NumPy

Introduction

Matrix multiplication helps combine data in rows and columns to find relationships. The @ operator makes this easy and clear.

When combining features and weights in machine learning models.

When transforming coordinates in graphics or physics.

When calculating total effects in economics or social sciences.

When working with multiple datasets that relate through matrix math.

Syntax

NumPy

result = matrix1 @ matrix2

The @ operator performs matrix multiplication, not element-wise multiplication.

Both matrices must have compatible shapes: columns in the first must equal rows in the second.

Examples

Multiply two 2x2 matrices using @.

NumPy

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

Multiply a 1x3 matrix by a 3x1 matrix to get a 1x1 matrix.

NumPy

import numpy as np
X = np.array([[1, 2, 3]])
Y = np.array([[4], [5], [6]])
Z = X @ Y

Sample Program

This program multiplies two 2x2 matrices using the @ operator and prints the input matrices and the result.

NumPy

import numpy as np

# Define two matrices
matrix1 = np.array([[2, 3], [4, 5]])
matrix2 = np.array([[6, 7], [8, 9]])

# Multiply using @ operator
result = matrix1 @ matrix2

print("Matrix 1:")
print(matrix1)
print("\nMatrix 2:")
print(matrix2)
print("\nResult of matrix1 @ matrix2:")
print(result)

Important Notes

The @ operator was introduced in Python 3.5 for matrix multiplication.

Using @ is clearer and less error-prone than using np.dot() or np.matmul().

Summary

The @ operator multiplies matrices when their shapes match.

It is simple and readable for matrix math in numpy.

Use it to combine data in rows and columns easily.

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