Adding and scaling

Matrices of the same order are added entry by entry: if A = [aᵢⱼ] and B = [bᵢⱼ] are both m × n, then A + B = [aᵢⱼ + bᵢⱼ], also m × n. Matrices of different orders cannot be added.

A scalar multiple kA multiplies every entry by k: kA = [k·aᵢⱼ]. Doubling a production table doubles every figure in it.

The negative is −A = (−1)A, and the difference is A − B = A + (−1)B, again entry by entry.

Addition keeps the familiar laws. It is commutative (the order of A and B does not matter) and associative (brackets can move). The zero matrix of the same order changes nothing when added, so it is the additive identity, and −A cancels A to give O, so it is the additive inverse.

Scaling distributes: k(A + B) = kA + kB and (k + l)A = kA + lA.

Worked example: with rows (1, 2, 3), (2, 3, 1) for A and rows (3, −1, 3), (−1, 0, 2) for B, the matrix 2A − B has rows (−1, 5, 3) and (5, 6, 0).

Matrix equations are solved like number equations. From X + Y = P and X − Y = Q, adding gives 2X = P + Q and subtracting gives 2Y = P − Q. With P = [[5, 2], [0, 9]] and Q = [[3, 6], [0, −1]], X = [[4, 4], [0, 4]] and Y = [[1, −2], [0, 5]].