Matrix Addition Rules

Linear Algebra (Matrices) · Axiom Academy

When two matrices can be added, how the sum is built entry by entry, and what scaling one by a number does to every entry at once. To add two matrices of the same size, you add the numbers that sit in the same position . The entry in row i , column j of the sum is the entry (i,j) of A plus the entry (i,j) of B — nothing crosses between positions. Each entry is the sum of the matching entries Entry-wise addition only makes sense when every position in A has a matching position in B . So two matrices can be added only if they have the same number of rows and the same number of columns . A matrix can be added to another , but never to a — there is no way to line their entries up. A plus a gives a . Every entry has a partner. A plus a has no entry-by-entry pairing. The sum does not exist. Multiplying a matrix by a single number c — a scalar — stretches every entry by the same factor . The entry (i,j) of cA is just c times the entry (i,j) of A . Slide the factor from 0 up to 3 and watch all four entries grow together, in lockstep. Because addition happens independently in each position, matrix addition inherits the familiar rules of ordinary number addition — applied entry by entry. A+B = B+A . Order does not matter in any position. (A+B)+C = A+(B+C) . Grouping does not matter. A+0 = A , where 0 is the all-zeros matrix of the same size. A+(-A) = 0 . The entrywise negative cancels A .

This is the written version of the interactive lesson above. See the full Linear Algebra (Matrices) course.