Matrix Addition and Multiplication Rules

Quick answer: Add matrices only when their dimensions match, adding corresponding entries. Multiply an m×n matrix by an n×p matrix using row-by-column dot products; the result has dimensions m×p.

Worked example

Let A = [[1,2],[3,4]] and B = [[2,0],[1,5]]. Find A + B and AB.

1. Entrywise addition gives A + B = [[3,2],[4,9]].
2. For AB, first row is [1×2 + 2×1, 1×0 + 2×5] = [4,10].
3. Second row is [3×2 + 4×1, 3×0 + 4×5] = [10,20].
4. Thus AB = [[4,10],[10,20]].

Check the result

BA = [[2,4],[16,22]], showing multiplication need not commute.

Common mistake to avoid

Matrix multiplication is not entrywise multiplication. The inner dimensions must agree even when the outer dimensions differ.

Try it yourself

1. What is the result size for (2×3)(3×4)? 2. Can a 2×3 matrix be added to a 3×2 matrix?

Answers and reasoning

1. 2×4. 2. No: addition requires exactly the same dimensions.

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