I have to challenge you with two questions that I gave up trying: 1. Is it true that the column/row space of $A^2$ is a subspace of the column/row space of $A$? If so, how? 2. Suppose I have a square matrix A of non-negative entries. Let B be another matrix obtained from A by replacing one or more (but not all) of the columns with the corresponding negatives as shown in the eg. When do the columns of $A^k$ and $B^k$ span the same vector spaces? [for k=1, clear!]
eg. only one column change, i.e. the last column of B is the negative of the last column of A
$A =\left(\begin{array}{cc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ \end{array}\right)$ , $B =\left(\begin{array}{cc} 1 & 2 & -3 \\ 4 & 5 & -6 \\ 7 & 8 & -9 \\ \end{array}\right)$
Thanks for your help!