Texed

Definition 02 · Matrices

Matrix addition

Combine two matrices of the same size by adding the entries that occupy matching positions.

A+B
Definition

The sum of two matrices with the same dimensions is the matrix formed by adding their corresponding entries.

The entrywise rule

Every entry keeps its row and column position. Addition is defined only when both matrices have exactly the same number of rows and columns.

Let A = [aij] and B = [bij].

A, B ∈ ℝm × n
A + B = [aij + bij]

(A + B)ij = aij + bij

A worked example

Select an entry in any matrix to follow that same position across the complete addition.

Matrix A
Matrix B
Sum
Position (1, 1): 1 + 4 = 5

Select any entry to trace its calculation

Key properties

Matrix addition inherits the familiar rules of ordinary number addition because each entry is added independently.

A + B = B + A

Commutative. Changing the order of the matrices does not change their sum.

(A + B) + C = A + (B + C)

Associative. Grouping does not matter when three equally sized matrices are added.

A + 0 = A

Additive identity. Adding the zero matrix of the same dimensions leaves every entry unchanged.

A + (−A) = 0

Additive inverse. A matrix and its negative cancel entry by entry.

Dimensions must match

Unlike multiplication, matrix addition never combines rows with columns. It simply pairs entries that already share an address.

Defined

(2 × 3) + (2 × 3)

Every entry has a partner in the same row and column.

Not defined

(2 × 3) + (3 × 2)

The shapes differ, so corresponding positions cannot be paired.

Common pitfall. Do not add only the diagonal or combine entire rows. Every position (i, j) in the first matrix is added only to position (i, j) in the second.