Matrix multiplication with @ operator in NumPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

What is the output of this matrix multiplication?

Given two numpy arrays A and B, what is the result of A @ B?

NumPy

import numpy as np
A = np.array([[1, 2], [3, 4]])
B = np.array([[2, 0], [1, 2]])
result = A @ B
print(result)

A

[[4 2]
 [7 8]]

B

[[4 4]
 [10 8]]

C

[[2 4]
 [3 8]]

D

[[2 0]
 [1 2]]

💡 Hint

Remember matrix multiplication sums the products of rows of A with columns of B.

Matrix multiplication with @ multiplies rows of A by columns of B and sums the products. For example, element (0,0) is 1*2 + 2*1 = 4.

❓ data_output

intermediate

What is the shape of the result after matrix multiplication?

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

NumPy

import numpy as np
A = np.zeros((3,4))
B = np.zeros((4,2))
result = A @ B
print(result.shape)

A (3, 4)

B (4, 4)

C (2, 3)

D (3, 2)

💡 Hint

The resulting matrix shape is rows of A by columns of B.

Matrix multiplication results in a matrix with the number of rows of the first matrix and the number of columns of the second matrix.

🔧 Debug

advanced

Why does this matrix multiplication raise an error?

What error will this code raise and why?

NumPy

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

A IndexError: index out of bounds

B TypeError: unsupported operand type(s) for @: 'list' and 'ndarray'

C ValueError: shapes (2,3) and (2,2) not aligned: 3 (dim 1) != 2 (dim 0)

D No error, outputs a (2,2) matrix

💡 Hint

Check if the inner dimensions of the matrices match for multiplication.

Matrix multiplication requires the number of columns in A to equal the number of rows in B. Here, A has 3 columns but B has 2 rows, so it raises a ValueError.

🚀 Application

advanced

Calculate the product of a matrix and its transpose

Given matrix M, what is the result of M @ M.T?

NumPy

import numpy as np
M = np.array([[1, 2], [3, 4], [5, 6]])
result = M @ M.T
print(result)

A

[[ 1 3 5]
 [ 2 4 6]]

B

[[ 5 11 17]
 [11 25 39]
 [17 39 61]]

C

[[ 1 2 3]
 [3 4 5]
 [5 6 7]]

D

[[ 5 11 17]
 [11 25 41]
 [17 39 61]]

💡 Hint

Multiplying a matrix by its transpose results in a symmetric matrix with dot products of rows.

M @ M.T computes dot products of rows of M with each other, resulting in a symmetric matrix with shape (3,3).

🧠 Conceptual

expert

Which statement about the @ operator in numpy is TRUE?

Select the correct statement about the @ operator for numpy arrays.

A The @ operator performs matrix multiplication and requires the inner dimensions to match.

B The @ operator performs element-wise multiplication of two numpy arrays.

C The @ operator can multiply any two numpy arrays regardless of their shapes.

D The @ operator is only used for multiplying numpy arrays with scalars.

💡 Hint

Think about how matrix multiplication works and what the @ operator does.

The @ operator performs matrix multiplication, which requires the number of columns in the first array to match the number of rows in the second array.

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