matrix addition

Matrix addition is the most natural thing imaginable: line up two same-shaped grids and add the numbers that sit in the same spot. It is exactly like adding two scorecards cell by matching cell.

Precisely: if A and B have the same dimensions, their sum A + B is the matrix whose entry in row i, column j is a_ij + b_ij. The shape is unchanged; only the contents combine, position by position.

There is one strict rule: the matrices must have identical dimensions, or the sum is simply undefined — there is no sensible way to add a 2-by-2 to a 2-by-3. When the shapes do match, addition is commutative and associative, just like ordinary numbers, so the order and grouping never matter.