Linear Algebra Rank of a Matrix Problem

[Heeyeyey]

Let A be a n x n matrix with complex elements. Prove that the a(k) array, with k ∈ ℕ, where a(k) = rank(A^(k + 1)) - rank(A^k), is monotonically increasing.

Thank you! :)

 

[Evgeny.Makarov]

This follows from $\operatorname {rank} (AB)\leq \min(\operatorname {rank} (A),\operatorname {rank} (B))$, if by "increasing" you mean "not decreasing".

 

[Heeyeyey]

Finally solved it using the Frobenius Inequality for the rank of a matrix. Thank you anyway!