Let a₁, a₂, ..., aₖ ∈ ℂ^(m×n) and let α ∈ ℂ. Prove by induction on k that for k ≥ 2:
a) a₁ + a₂ + ... + aₖ = α
b) αa₁ + αa₂ + ... + αaₖ = 0
c) a₁a₂ ... aₖ = α^k
d) α^(k-1)a₁a₂ ... aₖ = 0