The formula
(A ± B)ᵢⱼ = Aᵢⱼ ± Bᵢⱼ
What it means
Matrix addition works position by position: the element in row 2, column 3 of the result is the sum of the elements in row 2, column 3 of each matrix. Nothing mixes, which is why both matrices must have identical dimensions.
How to do it by hand
- Check both matrices have the same number of rows and columns
- Add or subtract each pair of elements in matching positions
- The result has the same size as the originals
- For subtraction, order matters: A − B is not B − A
What is worth knowing
Addition is the well-behaved matrix operation: it is commutative, associative, and has a neutral element in the zero matrix. Multiplication is none of those things, which is what makes linear algebra interesting and difficult. The trace, the sum of the diagonal, adds like the matrices do — the trace of a sum is the sum of the traces. The determinant does not: det(A + B) has no relation to det(A) + det(B), and expecting it to is one of the commonest early mistakes.