Lesson 8 of 13 · 7 min
Laws of multiplication
NCERT §3.4.6
A bill is orders times prices, and a term's spending is bills times weeks. Does it matter which product is worked out first?
The lesson in notes
In short
Associative law: (AB)C = A(BC) whenever both sides are defined. This is why A² = AA and A³ = AAA need no brackets.
Distributive laws: A(B + C) = AB + AC and (A + B)C = AC + BC, whenever defined. The side the common factor sits on must be kept.
Identity: for a square matrix A of order n, IA = AI = A, where I = Iₙ.
Scalars move freely: (kA)B = A(kB) = k(AB).
Because AB and BA can differ, (A + B)² = A² + AB + BA + B². It equals A² + 2AB + B² only when AB = BA. Likewise (A + B)(A − B) = A² − AB + BA − B².
Worked example: for A with rows (1, 2, 3), (3, −2, 1), (4, 2, 1), A² has rows (19, 4, 8), (1, 12, 8), (14, 6, 15) and A³ has rows (63, 46, 69), (69, −6, 23), (92, 46, 63). Then A³ − 23A − 40I = O.
A polynomial in A replaces the constant term c by cI: A² − 5A + 7 means A² − 5A + 7I. For A = [[3, 1], [−1, 2]], A² = [[8, 5], [−5, 3]] and A² − 5A + 7I = O.