Let $M$ be a $0-1$ matrix.

Here a matrix has one component means we can traverse from a matrix entry $(i,j)$ which is $1$ to any other one by moving step of $(i\pm1,j),(i,j\pm1),(i\pm1,j\pm1)$ where each step you take you step on another $1$.

Can every $0-1$ be converted to a matrix of one component by permutations of rows and columns?

What classes of matrices cannot have one component?

also posted: http://math.stackexchange.com/questions/1072461/connected-components-0-1-matrices

(Say I proved it for $n$ components merged to one. Now say I have $n+1$ components. If I move the first $n$ components by induction argument, the last $(n+1)^{st}$ component may split up in exponentially many. Am I wrong about this?)