A matrix is a rectangular, two-dimensional array of numbers. To multiply two matrices, match the first matrix’s column count to the second matrix’s row count; the product keeps the first matrix’s row count and the second matrix’s column count. Each output entry is the dot product of one row and one column.
What is a matrix?
A matrix is a two-dimensional arrangement of entries in rows and columns. Its shape is written as (rows, columns). For example, a matrix with three rows and two columns has shape 3 × 2, and in mathematical notation can be described as an element of ℝ3×2.
Mathematics commonly labels entries starting at 1: the entry in row i and column j is written aij. NumPy uses zero-based indexing instead, so the first row and first column are indexed with 0. Keep those conventions distinct when translating formulas into code.
When is a matrix product defined?
For A with shape m × n and B with shape n × p, the product AB is defined because A’s column count equals B’s row count. Its shape is m × p. The matching dimensions are called the inner dimensions; the dimensions that remain, m and p, determine the output.
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If the inner dimensions do not match, the product in that order is undefined. Matrix multiplication is not generally commutative: even when BA is defined, it need not have the same shape or values as AB.
How to calculate a matrix product
Consider these matrices, with their shapes shown first:
A (3 × 2) = [[1, 2], [3, 4], [5, 6]]
B (2 × 2) = [[7, 8], [2, 1]]
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The inner dimensions are both 2, so AB is defined and has shape 3 × 2. Multiply each row of A by each column of B using a dot product:
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- First row, second column: 1 × 8 + 2 × 1 = 10.
- Second row, first column: 3 × 7 + 4 × 2 = 29.
- Second row, second column: 3 × 8 + 4 × 1 = 28.
- Third row, first column: 5 × 7 + 6 × 2 = 47.
- Third row, second column: 5 × 8 + 6 × 1 = 46.
Thus AB = [[11, 10], [29, 28], [47, 46]]. Each result row corresponds to a row of A, and each result column corresponds to a column of B.
Matrix-vector and matrix-matrix products
Matrix times a vector
A matrix-vector product is the special case where the right operand has one column. For A shaped m × n and a column vector x shaped n × 1, Ax has shape m × 1. Each output value is the dot product of a row of A with x.
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There is another useful interpretation: the entries of x weight the columns of A, and Ax is the resulting linear combination of those columns. This view connects matrix multiplication to the idea of a linear transformation.
In NumPy, a one-dimensional array has shape (n,), not (n, 1) or (1, n). Consequently, multiplying an (m, n) array by an (n,) array with @ returns shape (m,), a one-dimensional result. To get a two-dimensional column result, reshape the vector to (n, 1) first.
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Matrix times a matrix
A matrix-matrix product can be understood as applying the matrix-vector operation to each column of the second matrix. If A is m × n and B is n × p, each of B’s p columns produces one output column, so AB is m × p.
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For example, multiplying a 3 × 2 matrix by a 2 × 3 matrix is defined and produces a 3 × 3 matrix. The inner 2s match; the outside dimensions, 3 and 3, give the result shape.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Matrix multiplication in Python with NumPy
NumPy’s @ operator performs matrix multiplication for two-dimensional arrays. The following example makes the operand and output shapes explicit:
import numpy as np
A = np.array([[1, 2],
[3, 4],
[5, 6]]) # shape (3, 2)
B = np.array([[7, 8],
[2, 1]]) # shape (2, 2)
C = A @ B # shape (3, 2)
print(C)
# [[11 10]
# [29 28]
# [47 46]]For a column-shaped matrix-vector result, use a two-dimensional column array:
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x = np.array([2, 3])
x_column = x.reshape(2, 1) # shape (2, 1)
y_column = A @ x_column # shape (3, 1)By contrast, A @ x with the original one-dimensional x returns shape (3,), not (3, 1). That distinction matters when later operations depend on whether an array is one-dimensional or explicitly a row or column matrix.
Why matrix products matter: covariance
Suppose a dataset has n observations in rows and variables in columns. Let X be the data matrix after centering each column (subtracting that variable’s mean from each observation). Then XᵀX is a variables-by-variables matrix whose entries collect products of pairs of centered variables across observations.
Dividing XᵀX by n − 1 gives the sample covariance matrix in this setup. Dividing by n gives the population-form covariance described in the example. The orientation matters: with observations in rows and variables in columns, transposing X before multiplying produces the covariance matrix across variables.
Continue learning
Hadrien Jean’s Essential Math for Data Science presents matrices and tensors, including matrix products, in a practical, code-supported context for data science and machine learning. The author’s introduction to matrices and matrix products is dated December 16, 2020, and is identified as a sample from the book. The author’s book page and O’Reilly catalog entry provide further information; check the intended edition and current retailer listing before choosing a copy.
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