ValueError: Shapes not aligned for matrix multiplication

When performing matrix multiplication in Python using numpy.dot() or the @ operator, you might encounter the error:

ValueError: shapes (m,n) and (p,q) not aligned for matrix multiplication

This error occurs when the number of columns in the first matrix does not match the number of rows in the second matrix.

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1. Understanding Matrix Multiplication Rules

For two matrices A (m × n) and B (p × q) to be multiplied:

🔹 Condition: The number of columns of A (n) must match the number of rows of B (p).
🔹 Resulting Matrix Shape: The resulting matrix will have the shape (m × q).

Valid Multiplication Example

If:

• Matrix A has shape (3 × 2) → 3 rows, 2 columns
• Matrix B has shape (2 × 4) → 2 rows, 4 columns
• Since A’s columns (2) match B’s rows (2), the multiplication is valid.
• The resulting matrix will have shape (3 × 4).

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2. Common Causes of Shape Misalignment

Mistake 1: Mismatched Dimensions

Incorrect Code:

import numpy as np

A = np.array([[1, 2, 3], [4, 5, 6]])  # Shape (2 × 3)
B = np.array([[7, 8], [9, 10]])       # Shape (2 × 2)

result = np.dot(A, B)  #  Error: (2×3) * (2×2) not possible

Output:

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

Problem:

• A (2 × 3) matrix cannot multiply with a (2 × 2) matrix.
• The number of columns in A (3) ≠ rows in B (2).

Fix: Ensure Matrix Shapes Are Compatible

B = np.array([[7, 8, 9], [10, 11, 12], [13, 14, 15]])  # Shape (3 × 3)
result = np.dot(A, B)  #  Now (2×3) * (3×3) is valid

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Mistake 2: Multiplying Vectors with Mismatched Shapes

Incorrect Code:

import numpy as np

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

result = np.dot(A, B)  #  Error: Shape mismatch

Output:

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

Problem:

• A (3,) vector cannot multiply with a (2,) vector.

Fix: Ensure One Vector is Reshaped to Match Dimensions

A = np.array([1, 2, 3]).reshape(3, 1)  # Convert to (3×1) matrix
B = np.array([4, 5, 6])                # Shape (3,)
result = np.dot(A.T, B)  #  (1×3) * (3×1) is valid

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Mistake 3: Using @ Operator Incorrectly

Incorrect Code:

import numpy as np

A = np.array([[1, 2], [3, 4]])  # Shape (2×2)
B = np.array([5, 6, 7])         # Shape (3,)

result = A @ B  #  Error: Shape mismatch

Output:

ValueError: matmul: Input operand 1 has a mismatch in its core dimension 0

Problem:

• A has 2 columns, but B has 3 elements → Shapes don’t align.

Fix: Reshape B Properly

B = np.array([5, 6]).reshape(2, 1)  # Convert to (2×1)
result = A @ B  #  Now (2×2) * (2×1) is valid

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Mistake 4: Transposing Instead of Reshaping

Incorrect Code:

import numpy as np

A = np.array([[1, 2, 3], [4, 5, 6]])  # Shape (2×3)
B = np.array([[7, 8, 9], [10, 11, 12]])  # Shape (2×3)

result = np.dot(A, B)  #  Error: Shape mismatch

Problem:

• Both matrices have (2 × 3) shape, but 3rd dimension (columns) must match 1st dimension (rows).

Fix: Use .T (Transpose) to Align Shapes

result = np.dot(A, B.T)  #  Now (2×3) * (3×2) is valid

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3. Best Practices to Avoid Shape Errors

1. Always Print Matrix Shapes

Before performing matrix multiplication, check matrix shapes using .shape:

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

print(A.shape)  # (2, 2)
print(B.shape)  # (2, 3)

if A.shape[1] == B.shape[0]:  # Columns of A must match rows of B
    result = np.dot(A, B)
else:
    print("Shape mismatch! Cannot multiply.")

2. Reshape Vectors Correctly

A = np.array([1, 2, 3]).reshape(3, 1)  # Convert to column vector (3×1)
B = np.array([4, 5, 6])  # Row vector (3,)

result = A.T @ B  #  (1×3) * (3,) = (1×3)

3. Use .T for Transposing Instead of .reshape()

A = np.array([[1, 2, 3], [4, 5, 6]])  # (2×3)
B = np.array([[7, 8, 9], [10, 11, 12]])  # (2×3)

result = np.dot(A, B.T)  #  Convert B to (3×2) to match A (2×3)

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4. Summary of Key Mistakes & Fixes

Mistake	Problem	Fix
Multiplying mismatched shapes	Columns of A ≠ Rows of B	Ensure A.shape[1] == B.shape[0]
Multiplying vectors incorrectly	Shape mismatch	Use .reshape() or .T
Using @ operator incorrectly	Wrong operand dimensions	Convert dimensions to match
Transposing incorrectly	Shape still misaligned	Use .T to fix alignment
Forgetting to print shapes before multiplication	Can’t debug errors	Use print(A.shape, B.shape) before multiplication