Medium

Matrix Addition and Subtraction

Simple Explanation

To add or subtract two matrices, add or subtract the entries in matching positions. This only works when both matrices have exactly the same order — you cannot add matrices of different sizes.

Why Do We Need It?

Matrix addition combines data stored in matrix form directly and simply, without needing to track individual values separately.

Formula

Matrix Addition and Subtraction

(A ± B)ᵢⱼ = Aᵢⱼ ± Bᵢⱼ

To add or subtract two matrices, add or subtract the entries that sit in the exact same position — this only works when both matrices have identical order (same number of rows and columns).

Aᵢⱼ: — the entry of matrix A in row i, column j
Bᵢⱼ: — the entry of matrix B in the same position

When to use it: Whenever you need to add or subtract two matrices of the same order.

Worked Example

Add two matrices

Find A + B, where A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]].

Why Does This Work?

Since a matrix is simply a labeled collection of individually-positioned numbers, adding two matrices of the same order is defined to mean adding the numbers that share the exact same label (position) — there is no other sensible way to combine two grids of numbers position-for-position.

Real-Life Example

Combining two months' sales figures

A company has one matrix of sales figures for January and another, same-shaped matrix for February.

Adding the two matrices gives the combined two-month sales for every product, entry by entry, in one operation.

Practice

Find A − B, where A = [[8, 3], [2, 6]] and B = [[5, 1], [4, 2]].

Medium

Common mistake

Trying to add matrices of different orders — matrix addition is undefined unless both matrices have exactly the same number of rows and the same number of columns.

Quick Review

(A±B)ᵢⱼ = Aᵢⱼ ± Bᵢⱼ.
Only defined for matrices of the exact same order.
Add/subtract entries in matching positions.