Rank of a matrix: Gaussian method

We create the $4\times4$ matrix: $$A=\left( \begin{array}{cccc} 1 & 0 & 1 & 0 \\ 0 & 2 & -1 & 2 \\ -1 & -1 & 1 & 1 \\ 0 & 1 & 1 & 3 \end{array} \right)$$ As we can see $r4=r1+r2+r3$.

Let's calculate the rank.

The first thing is to remove the linear row combination of others, that is, remove row $4$.

-Obviously there is nonempty submatrix $1\times1$ (all nonzero elements are).

-Are there any nonempty $2\times2$ submatrix? Yes $$\left| \begin{array}{cc} 1 & 0 \\ 0 & 2 \end{array} \right| = 2 \neq 0 $$

-Are there any nonempty $3\times3$ submatrixes? Yes $$\left| \begin{array}{ccc} 1 & 0 & 1 \\ 0 & 2 & -1 \\ -1 & -1 & 1 \end{array} \right| = -1 \neq 0 $$

$$\left| \begin{array}{ccc} 0 & 1 & 0 \\ 2 & -1 & 2 \\ -1 & 1 & 1 \end{array} \right| = -2-2=-4 \neq 0 $$

As the $4\times4$ matrix is empty (the row $4$ is a linear combination of the other $3$ rows) the order of the biggest nonempty square submatrix is $3$.

Therefore, rank$(A)=3$.