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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 $\mathrm{rank}(A)=\mathrm{rank}(A^k)$?

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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 $\begin{pmatrix}A \\ A^3 \end{pmatrix}$.

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  • $\begingroup$ Could You describe, what is the image of a matrix? $\endgroup$ – Jonny Jan 9 '13 at 11:49
  • $\begingroup$ The image of a matrix is the subspace generated by its columns. This should be clear if you think of matrices as linear functions. Are you aware of this concept? $\endgroup$ – Git Gud Jan 9 '13 at 13:55
  • $\begingroup$ @user1551: Of course, thank you! $\endgroup$ – Fredrik Meyer Jan 9 '13 at 16:11
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Hint: Prove that $\text{rank}\left(A^{k+1}\right)\leq \text{rank}\left(A^k\right)$, for all $k\in \mathbb{N}$.

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