The rank of a matrix is the number of linearly independent rows or columns in the matrix. To find the rank of a matrix, we can use Gaussian elimination.
In Gaussian elimination, we reduce the matrix to row echelon form. A row echelon form is a matrix in which all the rows below the main diagonal are zero, and the leading coefficient of each nonzero row is 1.
To reduce the matrix to row echelon form, we can use the following operations:
- Add or subtract a multiple of one row to another row.
- Multiply a row by a non-zero constant.
- Swap two rows.
Once we have reduced the matrix to row echelon form, the number of nonzero rows in the row echelon form is the rank of the original matrix.
In the given matrix, we can perform the following row operations:
- Subtract row 1 from row 2:
[\left[ {\begin{array}{*{20}{c}} 1&1&1 \ 0&-2&-1 \end{array}} \right]] - Add row 1 to row 3:
[\left[ {\begin{array}{*{20}{c}} 1&1&1 \ 0&-2&-1 \ 0&2&2 \end{array}} \right]]
We can see that the matrix is now in row echelon form. There are two nonzero rows in the row echelon form, so the rank of the original matrix is 2.
Therefore, the correct answer is $\boxed{\text{C}}$.