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 coefficients of the rows above the main diagonal are all 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 rank of the matrix is the number of non-zero rows in the row echelon form.
For the matrix $A$, we can reduce it to row echelon form as follows:
[\left[ {\text{A}} \right] = \left[ {\begin{array}{{20}{c}} 4&2&1&3 \ 6&3&4&7 \ 2&1&0&1 \end{array}} \right]\xrightarrow{R_2-\frac{3}{2}R_1\to R_2} \left[ {\begin{array}{{20}{c}} 4&2&1&3 \ 0&1&\frac{5}{2}&\frac{5}{2} \ 2&1&0&1 \end{array}} \right]\xrightarrow{\frac{1}{2}R_3\to R_3} \left[ {\begin{array}{*{20}{c}} 4&2&1&3 \ 0&1&\frac{5}{2}&\frac{5}{2} \ 0&0&-\frac{1}{2}&\frac{3}{2} \end{array}} \right]]
The row echelon form of the matrix $A$ has 3 non-zero rows, so the rank of the matrix $A$ is 3.
Therefore, the correct answer is $\boxed{\text{C}}$.