Powers and matrix equations

Powers of a square matrix are repeated products: A² = AA, A³ = A²A, and so on. Because multiplication is associative, Aᵐ Aⁿ = Aᵐ⁺ⁿ.

For the rotation-type matrix A = [[cos θ, sin θ], [−sin θ, cos θ]], induction gives Aⁿ = [[cos nθ, sin nθ], [−sin nθ, cos nθ]] for every natural number n: turning n times by θ is one turn by nθ.

A matrix with A² = A (idempotent) collapses powers: Aⁿ = A for all n ≥ 1, so (I + A)³ = I + 7A and hence (I + A)³ − 7A = I.

An unknown matrix is found by giving it letter entries of the right order and equating positions. For CD − AB = O with 2 × 2 matrices C = [[2, 5], [3, 8]], A = [[2, −1], [3, 4]] and B = [[5, 2], [7, 4]], AB = [[3, 0], [43, 22]] and the four equations give D = [[−191, −110], [77, 44]].

An equation like A² = kA − 2I fixes k by comparing one convenient entry and checking the others. For A = [[3, −2], [4, −2]], A² = [[1, −2], [4, −4]], and k = 1 works in every position.

Order is read before anything is multiplied: if the product of an unknown X with a 2 × 3 matrix is 2 × 3, then X must be 2 × 2.