rank of specified matrix

Question

I have a task to prove, that:

$$\mathrm{rank} \left(\begin{bmatrix}A &A^2 \\ A^3 & A^4 \end{bmatrix}\right)=\mathrm{rank}(A)$$

Please give me an advice. Is $\mathrm{rank}(A)=\mathrm{rank}(A^k)$?

Answer 1

First: No, the rank of $A$ and $A^k$ are not the same. For example, consider the matrix $A=\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$. Then $A^2=0$. So $A$ has rank $1$ but $A^2$ has rank zero.

To prove the proposition: The rank of a matrix is the maximum number of linearly independent columns. Hint: See that $A^2=A \cdot A$ and $A^4=A \cdot A^3$, so the right column have image contained in the image of the left column. Thus the image of the left matrix is the same as the image of $\begin{pmatrix}A \\ A^3 \end{pmatrix}$.

Answer 2

Hint: Prove that $rank(A^{k+1})\leq rank(A^k)$ for any $k\in \mathbb{N}$.