Linear Algebra Rank of a Matrix Problem: Prove that a(k) array is monotonically increasing

[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.

 

[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.

Sorry for reposting this here, but I was not 100% sure where I should post it and I really need a solution for this problem as I have tried to solve it quite a lot. Thank you!

 

[Heeyeyey]

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