Prove that raising a squared matrix to the power, keep its non linear rows

Question

I need to prove that raising a square matrix to any power, e.g. 3x3, will not change its non linear rows, meaning that in the end of gaussian elimination, I have the same matrix. Condition: at least one of the diagonal elements is not zero

e.g.

A= \begin{pmatrix} 1 & 2 &3 \\ 2& 4 & 6\\ 0 & 1 & 2 \end{pmatrix}

after gaussian elimination: \begin{pmatrix} 1 & 2 &3 \\ 0& 0 & 0\\ 0 & 1 & 2 \end{pmatrix} will still have same gaussian elimination result, even if I solve A^33 and then conduct gaussian elimination

Answer 1

If I understand the statement, it isn't true. Consider $$A=\begin{pmatrix}0&0&0\\0&0&0\\0&1&0\end{pmatrix}\\ A^2=\begin{pmatrix}0&0&0\\0&0&0\\0&0&0\end{pmatrix}$$