Types of matrices

Let's play a game. Here is a field with numbers:

$\begin{matrix} 1 & 0 & 2\\ 0 & 0 & 3\\ 4 & 5 & 6\\ \end{matrix}$

The field can be modified according to the following rules:

• you can rotate the rows to the left or right by one element. For example, like this:

$\begin{matrix} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9\\ \end{matrix} \longrightarrow \begin{matrix} 3 & 1 & 2\\ 4 & 5 & 6\\ 7 & 8 & 9\\ \end{matrix}$

$\begin{matrix} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9\\ \end{matrix} \longrightarrow \begin{matrix} 1 & 2 & 3\\ 5 & 6 & 4\\ 7 & 8 & 9\\ \end{matrix}$

• you can rotate the columns up or down as well. For example, like this:

$\begin{matrix} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9\\ \end{matrix} \longrightarrow \begin{matrix} 1 & 8 & 3\\ 4 & 2 & 6\\ 7 & 5 & 9\\ \end{matrix}$

$\begin{matrix} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 & 8 & 9\\ \end{matrix} \longrightarrow \begin{matrix} 1 & 2 & 6\\ 4 & 5 & 9\\ 7 & 8 & 3\\ \end{matrix}$

Your task is to specify in what minimum number of the above operations it is possible to bring the field to the form of an upper triangular matrix.