Matrix Operations in Python
1. Matrix Operations: Addition and Subtraction
Two matrices A and B of the same shape can be added:
123456789import numpy as np A = np.array([[1, 2], [5, 6]]) B = np.array([[3, 4], [7, 8]]) C = A + B print(f'C = {C}') # C = [[4, 6], [12, 14]]
2. Matrix Multiplication Rules
Matrix multiplication is not element-wise.
Rule: if A has shape (n,m) and B has shape (m,l), then the result has shape (n,l).
123456789101112131415# Example random matrix 3x2 A = np.random.rand(3, 2) print(f'A = {A}') # Example random matrix 2x4 B = np.random.rand(2, 4) print(f'B = {B}') # product shape (3, 4) product = np.dot(A, B) print(f'np.dot(A, B) = {product}') # or equivalently product = A @ B print(f'A @ B = {product}')
3. Transpose
Transpose flips rows and columns.
General rule: if A is (n×m), then AT is (m×n).
12345A = np.array([[1, 2, 3], [4, 5, 6]]) A_T = A.T # Transpose of A print(A_T)
4. Inverse of a Matrix
A matrix A has an inverse A−1 if:
A⋅A−1=Iwhere I is the identity matrix.
Not all matrices have inverses. A matrix must be square and full-rank.
Quiz
- What does
np.linalg.inv(A)do in Python when solving the system $Ax = b$?
A) Rotates vector $x$ B) Eliminates matrix $A$ C) Reverses the linear transformation applied by $A$ D) None of the above
Correct Answer: C
Correct Answer: B
12345678A = np.array([[1, 2], [3, 4]]) A_inv = np.linalg.inv(A) # Inverse of A print(A_inv) I = np.eye(2) # Identity matrix 2x2 print(assert np.allclose(A @ A_inv, I)) # Check if product equals identity
1. What does the following Python code print?
2. In Python's numpy, matrix multiplication can be performed between non-square matrices as long as their inner dimensions align.
3. What is the output of this Python code?
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1. Matrix Operations: Addition and Subtraction
Two matrices A and B of the same shape can be added:
123456789import numpy as np A = np.array([[1, 2], [5, 6]]) B = np.array([[3, 4], [7, 8]]) C = A + B print(f'C = {C}') # C = [[4, 6], [12, 14]]
2. Matrix Multiplication Rules
Matrix multiplication is not element-wise.
Rule: if A has shape (n,m) and B has shape (m,l), then the result has shape (n,l).
123456789101112131415# Example random matrix 3x2 A = np.random.rand(3, 2) print(f'A = {A}') # Example random matrix 2x4 B = np.random.rand(2, 4) print(f'B = {B}') # product shape (3, 4) product = np.dot(A, B) print(f'np.dot(A, B) = {product}') # or equivalently product = A @ B print(f'A @ B = {product}')
3. Transpose
Transpose flips rows and columns.
General rule: if A is (n×m), then AT is (m×n).
12345A = np.array([[1, 2, 3], [4, 5, 6]]) A_T = A.T # Transpose of A print(A_T)
4. Inverse of a Matrix
A matrix A has an inverse A−1 if:
A⋅A−1=Iwhere I is the identity matrix.
Not all matrices have inverses. A matrix must be square and full-rank.
Quiz
- What does
np.linalg.inv(A)do in Python when solving the system $Ax = b$?
A) Rotates vector $x$ B) Eliminates matrix $A$ C) Reverses the linear transformation applied by $A$ D) None of the above
Correct Answer: C
Correct Answer: B
12345678A = np.array([[1, 2], [3, 4]]) A_inv = np.linalg.inv(A) # Inverse of A print(A_inv) I = np.eye(2) # Identity matrix 2x2 print(assert np.allclose(A @ A_inv, I)) # Check if product equals identity
1. What does the following Python code print?
2. In Python's numpy, matrix multiplication can be performed between non-square matrices as long as their inner dimensions align.
3. What is the output of this Python code?
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