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 non-zero 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 the matrix is in row echelon form, the rank is the number of non-zero rows in the matrix.
For the matrix $M$, we can reduce it to row echelon form as follows:
[{\text{M}} = \left[ {\begin{array}{*{20}{c}} 5&{10}&{10} \ 1&0&2 \ 3&6&6 \end{array}} \right]]
Subtract 3 rows of the first row from the third row, we get
[{\text{M}} = \left[ {\begin{array}{*{20}{c}} 5&{10}&{10} \ 1&0&2 \ 0&6&-6 \end{array}} \right]]
Swap the second row with the first row, we get
[{\text{M}} = \left[ {\begin{array}{*{20}{c}} 1&0&2 \ 0&6&-6 \end{array}} \right]]
Divide the second row by 6, we get
[{\text{M}} = \left[ {\begin{array}{*{20}{c}} 1&0&2 \ 0&1&-1 \end{array}} \right]]
Since there are two non-zero rows in the row echelon form of the matrix, the rank of the matrix is 2.
Therefore, the correct answer is $\boxed{\text{C}}$.