Matrix Operations Summary

Linear Algebra (Matrices) · Axiom Academy

A recap of the three fundamental matrix operations — their dimension rules, the properties that carry over from ordinary arithmetic, and the ones that don't. Three operations, three roles: addition combines data, scalar multiplication scales it, and matrix multiplication transforms and composes it. Dimensions decide everything: addition needs identical dimensions; multiplication needs the inner dimensions to match, and the result is . Order matters for multiplication: in general — the single biggest break from number arithmetic. Some rules survive, many don't: associativity, distributivity, and identities carry over; commutativity of multiplication, free cancellation, and guaranteed inverses do not. Core Concept Notation & Dimensions A matrix is a rectangular array of numbers. Its dimension lists rows first, then columns; the entry a_ ij sits in row i , column j . When to use: always read off the dimension first — it determines which operations are even legal. Watch out for: rows before columns. A matrix has 2 rows and 3 columns, not the reverse. Two matrices are equal only when they share the same dimension and every corresponding entry agrees. There is no partial equality. When to use: setting matrices equal entry-by-entry to solve for unknowns. Watch out for: different shapes are never equal, even if they share values. Core Concept Addition & Subtraction

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