Matrix arithmetic
Multiplication: rows-times-columns. Not commutative. Identity matrix I acts like 1.
Matrix product AB defined when A is m×n and B is n×p; result is m×p. Element (i, j) = sum of products of A's row i with B's column j. AB ≠ BA in general.
Worked examples
(1 2; 3 4) × (5 6; 7 8) = (19 22; 43 50).
I·A = A·I = A; identity is the matrix multiplicative identity.
(AB)C = A(BC); associative but not commutative.
Frequently asked questions
Why non-commutative?
Geometric intuition: applying transformations in different orders gives different results.
(AB)<sup>T</sup>?
= BTAT. Order reverses on transpose.